A sequence of numbers Do, 11, 12,... is such that 10 = 4 and, for each n >1, In = Kon-1+3, where K is some fixed number such that 0 < K<1. Find an explicit expression for In in terms of n and K. ' 20 What is the limiting value of In as n tends to infinity? For which values of K will the sequence increase? For which will it decrease? For which will it be constant?
Answers
The explicit expression for In in terms of n and K is In = K^(n-1) * Do + 3 * (1 - K^(n-1)) / (1 - K). The limiting value of In as n tends to infinity is 3 / (1 - K). The sequence will increase for K = 1, decrease for 0 < K < 1, and be constant for K = 0.
The given sequence is defined recursively as In = Kon-1 + 3, where n > 1 and 0 < K < 1. We need to find an explicit expression for In in terms of n and K.
Let's write out the first few terms to observe a pattern:
I1 = K * Do + 3
I2 = K * I1 + 3 = K * (K * Do + 3) + 3 = K^2 * Do + 3K + 3
I3 = K * I2 + 3 = K * (K^2 * Do + 3K + 3) + 3 = K^3 * Do + 3K^2 + 3K + 3
From the pattern, we can see that each term In is obtained by multiplying the previous term by K and adding 3. Therefore, an explicit expression for In can be written as:
In = K^(n-1) * Do + 3 * (1 + K + K^2 + ... + K^(n-2))
Using the formula for the sum of a geometric series, we can simplify the expression inside the parentheses:
In = K^(n-1) * Do + 3 * (1 - K^(n-1)) / (1 - K)
Now, let's analyze the limiting value of In as n tends to infinity:
As n approaches infinity, the term K^(n-1) becomes smaller and approaches 0 since 0 < K < 1. Therefore, the limiting value of In is:
lim(n->∞) In = 3 / (1 - K)
Next, let's determine how the sequence behaves based on different values of K:
- If 0 < K < 1, the sequence will decrease since each term is multiplied by a number smaller than 1.
- If K = 0, the sequence will be constant, as each term is simply 3.
- If K = 1, the sequence will increase, as each term is equal to the previous term plus 3.
In summary, the explicit expression for In in terms of n and K is In = K^(n-1) * Do + 3 * (1 - K^(n-1)) / (1 - K). The limiting value of In as n tends to infinity is 3 / (1 - K). The sequence will increase for K = 1, decrease for 0 < K < 1, and be constant for K = 0.
Learn more about limiting value here:
https://brainly.com/question/29284611
#SPJ11
Related Questions
Unit 3 parallel and perpendicular lines Homework 2, please help quickly
Answers
Answer:
Step-by-step explanation:
Since, lines l and m are parallel and a transverse is intersecting these lines.
5). (9x + 2)° = 119° [Alternate intrior angles]
9x = 117 ⇒ x = 13
6). (12x - 8)° + 104° = 180°
12x = 180 - 96
x = \(\frac{84}{12}\) ⇒ x = 7
7). (5x + 7) = (8x - 71) [Alternate exterior angles]
8x - 5x = 71 + 7
3x = 78
x = 26
8). (4x - 7) = (7x - 61) [Corresponding angles]
7x - 4x = -7 + 61
3x = 54
x = 18
9). (9x + 25) = (13x - 19) [Corresponding angles]
13x - 9x = 25 + 19
4x = 44
x = 11
(13x - 19)° + (17y + 5)° = 180°[Linear pair of angles are supplementary]
(13×11) - 19 + 17y + 5 = 180
129 + 17y = 180
17y = 180 - 129
y = 3
10). (3x - 29) + (8y + 17) = 180 [linear pair of angles are supplementary]
3x + 8y = 180 + 12
3x + 8y = 192 -----(1)
(8y + 17) = (6x - 7) [Alternate exterior angles]
6x - 8y = 24
3x - 4y = 12 -----(2)
Equation (1) - equation (2)
(3x + 8y) - (3x - 4y) = 192 - 12
12y = 180
y = 15
From equation (1),
3x + 8(15) = 192
3x + 120 = 192
x = 24
11). (3x + 49)° = (7x - 23)° [Corresponding angles]
7x - 3x = 49 + 23
4x = 72 ⇒ x = 18
(11y - 1)° = (3x)° [Corresponding angles]
11y = 3×18 + 1
11y = 55 ⇒ y = 5
12). (5x - 38)° = (3x - 4)° [Corresponding angles]
5x - 3x = 38 - 4
2x = 34
x = 17
(7y - 20)° + (5x - 38)° + 90° = 180°
[Sum of interior angles of a triangle = 180°]
7y + 5x - 58 = 90
5x + 7y = 148
5×17 + 7y = 148
85 + 7y = 148
7y = 148 - 85
y = \(\frac{63}{7}=9\)
Angles can be congruent based n several theorems; some of these theorems are: corresponding angles, vertical angles, alternate exterior angles and several others.
The values of x and y are:
5. \(\mathbf{x = 13}\)6. \(\mathbf{x = 7}\)7. \(\mathbf{x= 26}\)8. \(\mathbf{x = 18}\)9. \(\mathbf{x = 11}\) and \(\mathbf{y = 7}\)10. \(\mathbf{x = 24}\) and \(\mathbf{y = 15}\)11. \(\mathbf{x = 18}\) and \(\mathbf{y =5}\)12. \(\mathbf{x = 17}\) and \(\mathbf{y=9}\)Question 5:
Angles (9x + 2) and 119 are alternate angles.
Alternate angles are equal. So, we have:
\(\mathbf{9x +2 = 119}\)
Subtract 2 from both sides
\(\mathbf{9x = 117}\)
Divide both sides by 9
\(\mathbf{x = 13}\)
Question 6:
Angles (12x - 8) and 104 are interior angles.
Interior angles add up to 180. So, we have:
\(\mathbf{12x -8 + 104 = 180}\)
Collect like terms
\(\mathbf{12x = 180 - 104 + 8}\)
\(\mathbf{12x = 84}\)
Divide both sides by 12
\(\mathbf{x = 7}\)
Question 7:
Angles (5x + 7) and (8x - 71) are alternate exterior angles.
Alternate exterior angles are equal. So, we have:
\(\mathbf{5x + 7 = 8x - 71}\)
Collect like terms
\(\mathbf{8x - 5x= 71 + 7}\)
\(\mathbf{3x= 78}\)
Divide both sides by 3
\(\mathbf{x= 26}\)
Question 8:
Angles (4x - 7) and (7x - 61) are corresponding angles.
Corresponding angles are equal. So, we have:
\(\mathbf{4x - 7 = 7x - 61}\)
Collect like terms
\(\mathbf{4x - 7x = 7 - 61}\)
\(\mathbf{- 3x = -54}\)
Divide both sides by -3
\(\mathbf{x = 18}\)
Question 9:
Angles (9x + 25) and (13x - 19) are corresponding angles.
Corresponding angles are equal. So, we have:
\(\mathbf{9x + 25 = 13x - 19}\)
Collect like terms
\(\mathbf{9x -13x = -25 - 19}\)
\(\mathbf{-4x = -44}\)
Divide both sides by -4
\(\mathbf{x = 11}\)
Angles (17y + 5) and (13x - 19) are supplementary angles.
So, we have:
\(\mathbf{17y + 5 = 13x - 19}\)
Substitute 11 for x
\(\mathbf{17y + 5 = 13\times 11 - 19}\)
\(\mathbf{17y + 5 = 124}\)
Subtract 5 from both sides
\(\mathbf{17y = 119}\)
Divide both sides by 17
\(\mathbf{y = 7}\)
Question 10:
Angles (3x - 29) and (6x - 7) add up to 180
So, we have:
\(\mathbf{3x -29 + 6x - 7 = 180}\)
Collect like terms
\(\mathbf{3x + 6x = 180 + 7 + 29}\)
\(\mathbf{9x = 216}\)
Divide both sides by 9
\(\mathbf{x = 24}\)
Angles (3x - 29) and (8y + 17) are supplementary angles.
So, we have:
\(\mathbf{3x - 29 + 8y + 17 = 180}\)
Substitute 24 for x
\(\mathbf{3 \times 24 - 29 + 8y + 17 = 180}\)
\(\mathbf{43 + 8y + 17 = 180}\)
Collect like terms
\(\mathbf{8y = 180 - 43 - 17}\)
\(\mathbf{8y = 120}\)
Divide both sides by 8
\(\mathbf{y = 15}\)
Question 11:
Angles (7x - 23) and (49 + 3x) are corresponding angles
So, we have:
\(\mathbf{7x - 23 = 49 + 3x}\)
Collect like terms
\(\mathbf{7x - 3x = 49 + 23}\)
\(\mathbf{4x = 72}\)
Divide both sides by 4
\(\mathbf{x = 18}\)
Angles 3x and (11y - 1) are corresponding angles.
So, we have:
\(\mathbf{3x = 11y - 1}\)
Substitute 18 for x
\(\mathbf{3 \times 18 = 11y - 1}\)
\(\mathbf{54 = 11y - 1}\)
Collect like terms
\(\mathbf{11y =54+ 1}\)
\(\mathbf{11y =55}\)
Divide both sides by 11
\(\mathbf{y =5}\)
Question 12:
Angles (5x - 38) and (3x - 4) are corresponding angles
So, we have:
\(\mathbf{5x - 38 = 3x - 4}\)
Collect like terms
\(\mathbf{5x - 3x = 38 - 4}\)
\(\mathbf{2x = 34}\)
Divide both sides by 2
\(\mathbf{x = 17}\)
Angles (7y - 20) and (5x - 38) are angles at the other legs of a right-angled triangle.
So, we have:
\(\mathbf{7y - 20 +5x - 38 = 90}\)
Substitute 17 for x
\(\mathbf{7y - 20 +5 \times 17 - 38 = 90}\)
\(\mathbf{7y+ 27 = 90}\)
Collect like terms
\(\mathbf{7y=- 27 +90}\)
\(\mathbf{7y=63}\)
Divide both sides by 7
\(\mathbf{y=9}\)
Read more about congruence angles at:
https://brainly.com/question/24623240
Tom sold t-shirts and hats at a festival. He made a $5 profit for each t-shirt he sold. He also made a profit of $40 from selling hats. If he made a total profit of $125, how many t-shirts did he sell?
Answers
The total number of t-shirts sold by Tom equals to 17 t-shirt.
What is a profit?Basically, a profit refers to the difference between the revenue that an entity has received from its outputs and the opportunity costs of its inputs.
We need to note that he made a total profit of $125. From here, we will proceed to use that data to solve other sections.
From the total profit of $125, he also made a profit of $40 from selling hats. That means that the T-shirt sales alone gives him a profit of $85 ($125 - $40).
Total quantity of t-shirt sold is computed as follows:
= Total profit from t-shirt / Profit per shirt
= $85 / $5
= 17 t-shirt
Read more about profit
brainly.com/question/1078746
#SPJ1
Answer number 4 this is due tomorrow
Answers
Answer:
The next mark should be at (3,-3)
Identify the vertex of the quadratic function:
f(x) = -(x - 2)2 – 3
Answers
Answer:
(2, -3)Step-by-step explanation:
\(f(x)=a(x-h)^2+k\) vertex form of equation with vertex (h, k)
\(\left \big{f(x)=a(x-h)^2+k}\atop \big{f(x)=-(x-2)^2-3}\right\}\implies a=-1\,,\ h=2\,,\ k=-3\)
So vertex of f(x) = - (x - 2)² - 3 is (2, -3)
Rendell cycles 42 km at an average speed of 18 km/hr. Find the time taken, giving your
answer as a fraction of an hour in its simplest form.
Answers
Answer:
2.3333333333333333333333333333333 sec
Hi Need Help on on this! Thank you so much! Will give a thumbs up!:)
The incremental fuel costs for two generating units 4 and B of a power plant are given by the following relations:
dFA/dPA=0.06 PA+ 11.4 dFy/dPa=0.07 Pa + 10 where P is the power in MW and F is the fuel cost in rupees per hour.
(a) Find the economic loading of the two units when the total load to be supplied by the power station is 150 MW.
(b) Find the net increase in fuel cost per hour if the load is equally shared by the two units.
Answers
(a) The economic loading of unit 4 and B are 11.54 MW and 138.46 MW, respectively.
To find the economic loading of the two units when the total load to be supplied by the power station is 150 MW, we need to minimize the total fuel cost. Let x be the power generated by unit 4 and y be the power generated by unit B. Then, we have:
x + y = 150 (total load)
The total fuel cost C is given by:
C = F4(x) + Fb(y)
where F4(x) and Fb(y) are the fuel costs for units 4 and B, respectively. Using the given relations, we have:
F4(x) = 0.06x^2 + 11.4x
Fb(y) = 0.07y^2 + 10y
Substituting x = 150 - y, we get:
C(y) = 0.06(150-y)^2 + 11.4(150-y) + 0.07y^2 + 10y
Expanding and simplifying, we get:
C(y) = 0.013y^2 - 3.6y + 1710
To minimize C(y), we take its derivative with respect to y and set it equal to zero:
dC/dy = 0.026y - 3.6 = 0
y = 138.46 MW
Substituting y back into x = 150 - y, we get:
x = 11.54 MW
(b) The negative value that is -1297.5 indicates that there is a net decrease in fuel cost per hour if the load is equally shared by the two units.
To find the net increase in fuel cost per hour if the load is equally shared by the two units, we need to calculate the fuel cost for each unit when they generate half of the total load (i.e., 75 MW). Using the given relations, we have:
F4(75) = 0.06(75)^2 + 11.4(75) = 1282.5
Fb(75) = 0.07(75)^2 + 10(75) = 1312.5
Therefore, the total fuel cost is:
C = F4(75) + Fb(75) = 2595
If the load is equally shared by the two units, each unit generates 75/2 = 37.5 MW. The fuel cost for each unit is:
F4(37.5) = 0.06(37.5)^2 + 11.4(37.5) = 641.25
Fb(37.5) = 0.07(37.5)^2 + 10(37.5) = 656.25
Therefore, the total fuel cost is:
C' = F4(37.5) + Fb(37.5) = 1297.5
The net increase in fuel cost per hour is:
ΔC = C' - C = 1297.5 - 2595 = -1297.5
To know more about derivative refer here:
https://brainly.com/question/29144258#
#SPJ11
9x7x9x9x7x9in index notation
Answers
What are the y-intercept and the asymptote of g(x) = 3x – 5? (0, –5); y = 3 (0, –2); y = 5 (0, –4); y = –5 (0, 5); y = –3
Answers
The y-intercept of the equation g(x) = 3^x - 5 is (0, -4) and the asymptote of the equation g(x) = 3^x - 5 is y = -5
How to determine the y-intercept?The equation of the function g(x) is given as:
g(x) = 3^x - 5
The y-intercept is a point on the graph where the value of x is 0
This is represented by x= 0 or (0, y)
This means that we substitute 0 for x in the above equation
So, we have:
g(0) = 3^0 - 5
Evaluate the exponent 3^0
g(0) = 1 - 5
Evaluate the difference of 1 and 5
g(0) = -4
Rewrite this point as
(0, -4)
This means that the y-intercept of the equation g(x) = 3^x - 5 is (0, -4)
How to determine the asymptote?The equation of the function g(x) is given as:
g(x) = 3^x - 5
The asymptote is a point on the graph where that is parallel to the graph
In the above equation, we have:
g(x) = 3^x - 5
Express the radical as 0
y = 0 - 5
Evaluate the difference of 0 and 5
y = -5
This means that the asymptote of the equation g(x) = 3^x - 5 is y = -5
Read more about asymptote at:
https://brainly.com/question/1851758
#SPJ1
2. The letters from the word MATH are placed in a hat. What is the probability that you select an A when you are choosing a letter without looking? A. What is asked in the problem? 8. What are the given numbers? What operation will you use? D What is the mathematical sentence? E What is the answer to the problem? V. A
Answers
The probability of getting M from the word MATH placed in hat is 0.25.
Probability is defined as the ratio of the number of favorable results to the total number of elements of an event.
P= number of favorable outcomes / total number of outcomes of an event
A. The problem is the probability of finding A
B. The given are MATH i.e. {M,A,T,H}
C. the operation used will be selecting a word from hat of word MATH
D. the mathematical statement from the probability of finding M will be
P(getting M from word MATH)
E. the answer to the problem will be calculated as
no. of element in the hat=4 i.e. {M,A,T,H}
no. of favorable outcomes= 1 i.e. {A}
P(getting A from word MATH) = number of favorable outcomes/ number of total outcomes
⇒ P(getting A from word MATH) = 1/4
Learn more about probability
here: https://brainly.com/question/251701
#SPJ10
determine the slope of the line passing through the given pair of points
through (-4,3) and (0,-4)
Answers
Explanation:
\(g(x) = x - 3\)\(f(x) = 2x^{3} \)Find f(x)Find g(4)Find (f+g)(x)Find f(0)+g(2)Find f(3)+g(0)Find f(-1) -f(1)+g(2)
Answers
g(x) = x - 3
f(x) = 2x³
to find g(4), replace x = 4 in g(x) formula, as follows:
g(4) = 4 - 3 = 1
(f+g)(x) = f(x) + g(x) = 2x³ + x - 3
f(0)+g(2) = 2(0)³ + (2) - 3 = 2 - 3 = -1
f(3)+g(0) = 2(3)³ + (0) - 2 = 2(27) - 2 = 54 - 2 = 52
f(-1) -f(1)+g(2) = 2(-1)³ - 2(1)³ + 2 - 3 = 2(-1) - 2(1) + 2 - 3 = -2 - 2 + 2 - 3 = -5
Write the decimal 2. 6 as a simplified fraction
Answers
The value of the decimal 2.6 is 13/50 under the condition that it needs to be in a form of simplified fraction.
Now to write the decimal 2.6 as a fraction, we to apply the following steps
The decimal's number of places must be counted. In this case, there is one decimal place.
With a denominator of 1 and as many zeros as there are decimal places, we have to write the fraction with the decimal as the numerator.
Then we get, for 2.6
2.6 = 2.6/10
Applying simplification to the given fraction by dividing both the numerator and denominator by their greatest common factor.
For this case, the greatest common factor of 26 and 10 is 2,
= 2.6/10
= 26/100
= 13/50
Hence, the decimal 2.6 as a simplified fraction is 13/50.
To learn more about fractions,
https://brainly.com/question/601741
#SPJ4
Pablo owes his sister $9. His sister Nina owes their mother three times that amount. What number on the number line represents Nina’s debt?
Answers
Answer:
Step-by-step explanation:
Nina's debt = 3 * 9 = $ 27
Debt is represented using negative sign.This represents (-27) on the number line.
Answer:(-27)
Step-by-step explanation:
i got it right on edge 2022
A 500 kg bridge is 25 meters long, and supported by two pillars. One pillar is 2 meters from the left end, the other is 5 meters from the right end. A 200 kg horse is standing on the bridge 6 meters from the left end. A 50 kg jockey is standing on the bridge 1 meter from the right end. Determine the amount of weight supported by each pillar (left, right).
Answers
To determine the amount of weight supported by each pillar, we need to first calculate the total weight of the bridge and everything on it.
Total weight = weight of bridge + weight of horse + weight of jockey
Total weight = 500 kg + 200 kg + 50 kg
Total weight = 750 kg
Next, we need to calculate the weight distribution on the bridge. We can do this by finding the center of mass of the system.
Center of mass = (weight of bridge x distance to center of bridge) + (weight of horse x distance to horse) + (weight of jockey x distance to jockey) / total weight
Center of mass = (500 kg x 12.5 m) + (200 kg x 6 m) + (50 kg x 24 m) / 750 kg
Center of mass = 9.47 m from the left end of the bridge
Now we can use the principle of moments to find the weight supported by each pillar.
Anti-clockwise moments = clockwise moments
Weight supported by left pillar x distance to left pillar = (750 kg x 9.47 m) - (200 kg x 3.47 m) - (50 kg x 23.47 m)
Weight supported by left pillar x distance to left pillar = 4662.5 kgm
Weight supported by right pillar x distance to right pillar = (200 kg x 16.53 m) + (50 kg x 24.53 m) - (750 kg x 15.53 m)
Weight supported by right pillar x distance to right pillar = 4662.5 kgm
Solving for each weight, we get:
Weight supported by left pillar = 330.5 kg
Weight supported by right pillar = 419.5 kg
Therefore, the left pillar supports 330.5 kg and the right pillar supports 419.5 kg of weight.
To determine the amount of weight supported by each pillar, we can use the principle of moments (torque). First, let's calculate the total weight acting on the bridge.
Total weight = Bridge weight + Horse weight + Jockey weight = 500 kg + 200 kg + 50 kg = 750 kg
Next, let's find the position of the center of mass of the system. We'll assume the bridge weight is evenly distributed along its length:
Center of mass = [(500 kg * 12.5 m) + (200 kg * 6 m) + (50 kg * 24 m)] / 750 kg ≈ 11.67 m from the left end
Now, we'll set up the equation for moments about the left pillar (counter-clockwise positive):
Moment_left - Moment_right = 0
(750 kg * 11.67 m) - (W_right * (25 m - 5 m)) = 0
Solve for W_right:
W_right ≈ 351 kg
Now we'll find the weight supported by the left pillar by subtracting W_right from the total weight:
W_left = 750 kg - W_right ≈ 399 kg
So, the left pillar supports approximately 399 kg, and the right pillar supports approximately 351 kg.
Learn more about arithmetic here: brainly.com/question/11559160
#SPJ11
Show that the convolution formula is unchanged if the input x and impulse response h are swapped, i.e. that ∫
−[infinity]
[infinity]
x(τ)h(t−τ)dτ=∫
−[infinity]
[infinity]
h(τ)x(t−τ)dτ (This symmetry between x and h allows us to write this as " x(t)∗h(t)
′′
.)
Answers
The expression obtained is identical to the original convolution integral, we have shown that the convolution formula remains unchanged if the input x and impulse response h are swapped. This symmetry between x and h allows us to write it as "x(t) * h(t)".
To show that the convolution formula is unchanged when the input x and impulse response h are swapped, we need to prove the commutativity of convolution.
Let's consider the convolution integral:
y(t) = ∫[−∞\(]^[∞]\) x(τ)h(t - τ) dτ
Now, we will interchange the roles of x and h, and rewrite the convolution integral as:
y(t) = ∫[−∞\(]^[∞]\) h(τ)x(t - τ) dτ
By comparing the two expressions, we can observe that the only difference is the swapping of x and h in the integrand.
To prove the symmetry and commutativity of convolution, we can perform a change of variable in the second integral:
Let τ' = t - τ
The limits of integration remain the same, as the integral is taken over the entire real line. Differentiating τ' with respect to τ gives dτ' = -dτ.
Substituting these into the second integral, we have:
y(t) = ∫[−∞\(]^[∞]\) h(τ')x(t - τ') (-dτ')
Notice that the limits of integration do not change, as they are independent of the variable of integration.
Now, let's reverse the order of integration by changing the sign of dτ':
y(t) = ∫[∞\(]^[−∞\)] h(τ')x(t - τ') dτ'
We can rename the dummy variable of integration back to τ:
y(t) = ∫[−∞\(]^[∞\)] h(τ)x(t - τ) dτ
To know more about convolution formula refer to-
https://brainly.com/question/32650611
#SPJ11
Complete Question
To show that the convolution formula is unchanged when the input function x(t) and impulse response function h(t) are swapped, we need to prove that:
∫[−∞]^[∞] x(τ)h(t - τ) dτ = ∫[−∞]^[∞] h(τ)x(t - τ) dτ
This symmetry between x and h allows us to write the convolution as "x(t) * h(t)".
HELP ME Pleasesssss??I need this ASAP please helpp me with this 3 Questions :)
1)A cylindrical container of radius 4cm and height 15cm is half filled with water.Find the volume of water in the container .Give the answer correct to 3 significant figures.
2)A can is a cylinder of radius 3.4cm . It is completely filled with 233cm³ of orange juice. Find the height of the can.Give the answer correct to 3 significant figures.
3)A lampshade is in the form of cylinder of radius 11cm .It is open at both ends.The total surface area of lampshade is 1200cm² . Find its height .Give your answer correct to 3 significant figures.
Answers
Answer:
753.6 cm³
21.825 cm
34.742 cm
Step-by-step explanation:
1) V=πr²h
r= 4 cm, h= 15 cm
V= 3.14*4²*15= 753.6 cm³
2) V=πr²h
r= 3.4 cm, V= 233 cm³, h=?
h= V/πr²= 233/(3.14*3.4)= 21.825 cm
3) A=2πrh (not including base surface as it is open ended)
r= 11 cm, A= 1200 cm², h=?
h= A/(2πr)= 1200/(3.14*11)= 34.742 cm
13. The students in charge of the class booth at a carnival would like to earn $3 for every item they sell. They spent $55 for the materials to make the items. Solve the inequality 3x – 55 > 65, which represents how many items they need to sell to make a profit of at least $65.
Answers
They need to sell more than 4 items to make a profit of at least $65
Inequality shows the relationship between non equal number or expressions. Common signs used in inequality is the greater than (>), greater than or equal to (≥), less than (<), less than or equal to (≤).
Let x represent the number of item sold. Since $3 is earned per item, hence the total money earned is:
Total money earned = 3x
They spent $55 and also need to make a profit of at least $65, hence:
3x – 55 > 65
3x > 120
x > 4
Therefore they need to sell more than 4 items
Find out more at: https://brainly.com/question/15137133
Frank bought n oranges. Carlos bought 8 fewer oranges than Frank. In terms of n, how many oranges did Carlos buy?
A. n + 8
B. n - 8
C. 8 - n
D. 8 - 8n
Answers
The number of oranges that Carlos bought is n - 8. n is variable.
What is an expression?
Mathematical expressions consist of at least two numbers or variables, at least one math operation, and a sentence. It's possible to multiply, divide, add, or subtract with this mathematical operation. An expression's structure is as follows: Expression: (Math Operator, Number/Variable, Math Operator).
The alphabetic character that expresses a numerical value or a number is known as a variable in mathematics. A variable is used to represent an unknown quantity in algebraic equations.
Any alphabet from a to z can be used for these variables. Most frequently, the variables "a," "b," "c," "x," "y," and "z" are used in equations.
Given that Frank bought n oranges. Carlos bought 8 fewer oranges than Frank.
The number of oranges that Frank bought is more than the number of oranges that Carlos.
The number of oranges that Carlos bought is n - 8.
To learn more about algebraic expression, click on the below link:
https://brainly.com/question/5829849
#SPJ1
Answer:
b just took the test and i put c trust me it is b
Step-by-step explanation:
Max is spending the afternoon at the ice rink with friends. it costs $7.00 to rent skates for a day, and $4.00 for each hour of skating. He has $20.00 to spend at the ice rink. Which inequality shows the number of hours, h, Max can skate if he also has to rent skates?
Answers
Answer:
20 greater than or equal to 4x + 7
Step-by-step explanation:
hope this helps!
Which of the following is true about the curve x^2 - xy + y^2 = 3 at the point (2,1)?
all of these are different answers, only one can be right.
a: dy/dx exists at (2,1) but there is no tangent line at that point
b; dy/dx exists at (2,1) , and the tangent line at that point is horizontal
c; dy/dx exists at (2,1), and the tangent line at that point is neither horizontal nor vertical
d: dy/dx does no exists at (2,1) and the tangent line at that point is vertical.
Answers
The correct answer is option C. At the point (2,1) on the curve x² - xy + y²= 3, the derivative dy/dx exists, and the tangent line at that point is neither horizontal nor vertical.
To determine the correct option, we need to analyze the properties of the curve x² - xy + y² = 3 at the point (2,1). First, let's find the derivative dy/dx. Taking the derivative of the given equation implicitly with respect to x, we get:
2x - y - x(dy/dx) + 2y(dy/dx) = 0
Rearranging the terms, we have:
dy/dx = (2x - y) / (x - 2y)
Now, substituting the values x = 2 and y = 1 into the expression for dy/dx, we can determine its value at the point (2,1):
dy/dx = (2(2) - 1) / (2 - 2(1)) = 3 / 0
Since the denominator is zero, dy/dx is undefined at (2,1). Therefore, option D, stating that dy/dx does not exist at (2,1) and the tangent line at that point is vertical, is incorrect.
However, we can still determine the nature of the tangent line at (2,1). Although the derivative is undefined, it is still possible for the tangent line to exist and have a defined slope. In this case, the tangent line would be neither horizontal nor vertical. Therefore, option C, which states that dy/dx exists at (2,1) and the tangent line at that point is neither horizontal nor vertical, is the correct answer.
Learn more about Tangent:
brainly.com/question/10053881
#SPJ11
* 11. Is the following proposition true or false? Justify your conclusion. For all integers a, b, and c, if gcd(a, b) = 1 and c | (a + b), then ged(a, c) = 1 and gcd(b, c) = 1.
Answers
The statement "For all integers a, b, and c, if gcd(a, b) = 1 and c | (a + b), then gcd(a, c) = 1 and gcd(b, c) = 1" is TRUE for all integers a, b, and c.
How to justify your conclusion that the proposition is true?The proof is required to demonstrate that the proposition is true.
Consider the following two statements:
If gcd(a, b) = 1, then there exists x and y, integers, such that ax + by = 1.
[Euclid's Lemma]a and b are coprime (that is, gcd(a, b) = 1) if and only if there exist integers x and y such that ax + by = 1.
[Bezout's Theorem]Here are the steps for proving that the proposition is true:
If gcd(a, b) = 1, then there exist integers x and y such that ax + by = 1.
Multiplying the previous equation by c yields cax + cby = c, which is equivalent to c | (a + b).
We must now demonstrate that if c | (a + b), then gcd(a, c) = 1 and gcd(b, c) = 1.
To accomplish this, suppose d is a common divisor of a and c.
Then, there exist integers x and y such that a = dx and c = dy.
Therefore, c | (a + b) means that dy | dx + b, which simplifies to d | b.
Thus, gcd(a, c) = d and d | b, so d is a common divisor of b and c.
The greatest common divisor of b and c is gcd(b, c), which is no more than d because d is a common divisor.
Similarly, gcd(b, c) is a divisor of a and c if d is a divisor of b and c.
If gcd(a, b) = 1 and c | (a + b), then gcd(a, c) = 1 and gcd(b, c) = 1.
This is true because of Euclid's lemma, which states that if a common divisor d divides two integers a and b, then it also divides their linear combination.
Therefore, gcd(a, c) and gcd(b, c) divide gcd(a, b) = 1, which means they are also equal to 1.
The proof is now complete.
The statement is true for all integers a, b, and c.
To know more about common divisor visit:
https://brainly.com/question/27962046
#SPJ11
A stack of Magic cards consists of 4 land cards and 6 creature cards. You draw 4 cards from the shuffled stack of 10 cards. Find the probability that you pull 4 creature cards. (give answer as a fraction or a decimal to 3 decimal places)
Find the probability that you pull 3 land cards and 1 creature card.
(give answer as a fraction or a decimal to 3 decimal places)
Answers
Answer:
1/14 ; 1/35
Step-by-step explanation:
A
6/10 x 5/9 x 4/8 x 3/7
3/5 x 5/9 x 1/2 x 3/7
1 x 1/3 x 3/14
1/14
B
4/10 x 3/9 x 2/8 x 6/7
2/5 x 1/3 x 1/4 x 6/7
2/15 x 3/14
1/5 x 1/7
1/35
The probability that 4 creature cards is pulled is 1/14 and the probability that 3 land cards and 1 creature card is pulled is 4/35
What is probability?Probability is a measure of the likelihood of an event occurring. The probability formula is defined as the possibility of an event happening is equal to the ratio of the number of favorable outcomes and the total number of outcomes. Probability of event to happen P(E) = Number of favorable outcomes/Total Number of outcomes.
Given here there are 4 land cards and 6 creature cards
thus selecting four cards at random from 10 cards is ¹⁰C₄=210
and the event of selecting 4 creature cards from a total of 6 cards is ⁶C₄ =15
Therefore probability of pulling 4 creature cards is = 15/210
= 1/14
The probability of pulling 3 land cards and 1 creature cards is
= ⁴C₃ × ⁶C₁ / ¹⁰C₄
= 4/35
Hence 1/14 and 4/35 are the respective probabilities.
Learn more about probability here:
https://brainly.com/question/30034780
#SPJ2
Find the maximum rate of change of f at the given point and the direction in which it occurs.
F(x, y, z) = (8x + 5y)/z
(5, 6, -1)
maximum rate of change
direction vector
Answers
The direction in which the maximum rate of change of f occurs at the point (5, 6, -1) is approximately (-0.17, -0.11, -0.98).
To find the maximum rate of change of the function f at the given point (5, 6, -1) and the direction in which it occurs, we can calculate the gradient of f at that point.
The gradient vector represents the direction of maximum increase of the function, and its magnitude represents the rate of change in that direction.
The gradient vector (∇f) of f(x, y, z) = (8x + 5y)/z can be found by taking the partial derivatives with respect to each variable:
∂f/∂x = 8/z
∂f/∂y = 5/z
∂f/∂z = -(8x + 5y)/z^2
Evaluated at the point (5, 6, -1), we have:
∂f/∂x = 8/(-1) = -8
∂f/∂y = 5/(-1) = -5
∂f/∂z = -((8(5) + 5(6))/(-1)^2) = -46
So, the gradient vector (∇f) at the point (5, 6, -1) is (-8, -5, -46).
The maximum rate of change of f at this point is given by the magnitude of the gradient vector:
|∇f| = √((-8)^2 + (-5)^2 + (-46)^2) = √(64 + 25 + 2116) = √2205 = 47.
Therefore, the maximum rate of change of f at the point (5, 6, -1) is 47.
To determine the direction in which this maximum rate of change occurs, we normalize the gradient vector by dividing it by its magnitude:
Direction vector = (∇f) / |∇f| = (-8/47, -5/47, -46/47).
Hence, the direction in which the maximum rate of change of f occurs at the point (5, 6, -1) is approximately (-0.17, -0.11, -0.98).
Learn more about direction from
https://brainly.com/question/29248951
#SPJ11
at the local pizza Parlor Game tickets can be trader for small toys. The rate is 10 tickets for 4 a small toys. If Meg 155 tickets playing basketball, for how many small twist can she treat her tickets?
Answers
why is 7a+4−2a is equivalent to 7a+-2a+4?
Answers
Answer:
communitive property it doen't matter which way you put it the equation will still be the same.
Step-by-step explanation:
7a+-2a+4
7a+4−2a
Candy choices at the movie theater are an example of which type of data? Statistic Extreme Qualitative Quantitative
Answers
The example of choosing candy falls under the dignified criteria of Qualitative data. Then the correct option required is Option B.
Qualitative data refers to data that is considered descriptive and conceptual which are collected through questionnaires, observation, and interviews. candy choices in the given question fall under the description of Qualitative data. Due to the following reasons
It is non-numerical and readily describes the characteristics of objects, people, places, etc.No calculations are included for choosing a particular option that will in conclusion help in determining the better candy.No involvement of 2nd person's preferences or choices to find a suitable candy.No limiting variables to hold the imagination, are required to select the perfect candy.To learn more about Qualitative data,
https://brainly.com/question/8064831
#SPJ4
I need help, can someone please answer this for me please
Answers
\( \large \tt \: m = - \frac{3}{7} \)
Detailed solution is attached!!~
Answer:
slope = - \(\frac{3}{7}\)
Step-by-step explanation:
Calculate the slope m using the slope formula
m = \(\frac{y_{2}-y_{1} }{x_{2}-x_{1} }\)
with (x₁, y₁ ) = (2, \(\frac{2}{7}\) ) and (x₂, y₂ ) = (1, \(\frac{5}{7}\) )
m = \(\frac{\frac{5}{7}-\frac{2}{7} }{1-2}\) = \(\frac{\frac{3}{7} }{-1}\) = - \(\frac{3}{7}\)
Ms.Liu has $60 to buy candy and soda at the movies theater for her family. Candy cost $6 per box and soda cost $4 per cup .Write an algebraic expression that can be used to find the amount of money mr.Liu will have left if she buys a number of boxes of candy (c)and cups of soda (s).
Answers
Given :
Ms. Liu has $60 to buy candy and soda at the movies theatre for her family.
Candy cost $6 per box and soda cost $4 per cup.
To Find :
Write an algebraic expression that can be used to find the amount of money ms. Liu will have left if she buys a number of boxes of candy (c) and cups of soda (s).
Solution :
Let, number of candy and soda are c and s.
So, their price is :
Price = 6c + 4s
Amount left = 60 - ( 6c + 4s )
Amount left = 60 - 6c -4s
Hence, this is the required solution.
\(\sqrt{25} is an irrational
Answers
Answer:
Is Square Root of 25 Rational or Irrational?
Step-by-step explanation:
A rational number can be expressed in the form of p/q. Because √25 = 5 and 5 can be written in the form of a fraction 5/1. It proves that √25 is rational.
The answer is:
⇨ √25 is a rational numberWork/explanation:
What are rational numbers?
Rational numbers are integers and fractions.
Irrational numbers are numbers that cannot be expressed as fractions, such as π.
Now, \(\bf{\sqrt{25}}\) can be simplified to 5 or -5; both of which are rational numbers.
Hence, √25 is rational.10. One contestant on a game show has
1,500 points and another contestant has
-250 points. What is the difference
between the scores of the contestants?
Answers
it's the difference between 1500 and -250, so simply add the two.
think about it like this:
one number is lower than zero, so you already have the 1500 minimum for a difference. but it isn't bang on zero, it's 250 less than that. so the answer is 1750.