If x is increasing at a
rate of 2 units per second,
find the rate of change of theta
at the instant when x= 12 units.
Answers
Answer:
\(\frac{d\theta}{dt}=-\frac{2}{5}\) at \(x=12\)
Step-by-step explanation:
\(\frac{dx}{dt}=2\)
\(\frac{d\theta}{dt}=?\)
\(x=12\)
\(cos(\theta)=\frac{adjacent}{hypotenuse}\)
\(cos(\theta)=\frac{x}{13}\)
\(\frac{d}{dt}cos(\theta)=\frac{d}{dt}\frac{x}{13}\)
\(-sin(\theta)\frac{d\theta}{dt}=\frac{1}{13}\frac{dx}{dt}\)
\(-sin(\theta)\frac{d\theta}{dt}=\frac{1}{13}(2)\)
\(-sin(\theta)\frac{d\theta}{dt}=\frac{2}{13}\)
\(\frac{d\theta}{dt}=\frac{2}{-13sin(\theta)}\)
\(cos(\theta)=\frac{x}{13}\)
\(cos(\theta)=\frac{12}{13}\)
\(\theta=cos^{-1}(\frac{12}{13})\)
\(\frac{d\theta}{dt}=\frac{2}{-13sin(\theta)}\)
\(\frac{d\theta}{dt}=\frac{2}{-13sin(cos^{-1}(\frac{12}{13}))}\)
\(\frac{d\theta}{dt}=\frac{2}{-13(\frac{5}{13})}\)
\(\frac{d\theta}{dt}=\frac{2}{-5}\)
\(\frac{d\theta}{dt}=-\frac{2}{5}\)
The value of θ is given by the inverse sine function, from which the rate of
change of θ with respect to x can be derived.
The rate of change of θ at the instant when x = 12 units is -0.4 rad/sReasons:
\(\displaystyle \frac{dx}{dt} = \mathbf{2 \ units \ per \ second}\)
\(\displaystyle cos(\theta) = \frac{x}{13}\)
\(\displaystyle \theta = arccos \left(\frac{x}{13}\right)\)
\(\displaystyle \frac{d}{dx} \theta = \frac{d\left(arccos \left(\frac{x}{13}\right)\right)}{dx} = \mathbf{\frac{\sqrt{169-x^2} }{x^2-169}}\)
Using chain rule of differentiation, we have;
\(\displaystyle \frac{d\theta}{dt} = \mathbf{ \frac{d\theta}{dx} \times \frac{dx}{dt}}\)
Therefore;
\(\displaystyle \frac{d\theta}{dt} =\frac{\sqrt{169-x^2} }{x^2-169}\times \frac{dx}{dt} = \mathbf{\frac{\sqrt{169-x^2} }{x^2-169}\times 2}\)
When x = 12, we get;
\(\displaystyle \frac{d\theta}{dt} =\frac{\sqrt{169-12^2} }{12^2-169}\times 2 = -\frac{2}{5} = -0.4\)
The rate of change of the angle, θ, with time at the instant when x = 12 is -0.4 rad/s
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Related Questions
what is 21.25 rounded to the nearest whole number
Answers
Answer:21
Step-by-step explanation:
-3/4 x 2/5 =? IN SIMPLEST FORM!!!! I mark BRAINLIESTTTTTT and im giving 35 POINTSSS PLEASE HELP ASAP
Answers
Answer:
\(-\frac{3}{10}\)
Step-by-step explanation:
4 × 5 = 20-3 × 2 = -6Put -6 on top of 20: \(-\frac{6}{20}\) Simplify by dividing each side by 2. It should now look like this: \(-\frac{3}{10}\)I hope this helps!
Answer:
-0.75 x 0.4= -0.3 or -3/10
Step-by-step explanation:
-3/4 translates to -0.75 and 2/5 translates to 0.4
please help with question b c d
Answers
If the variance of the data values in a sample is 289, what is the standard
deviation of the data values?
O A. 17
OB. 19
OC. 18
O D. 16
Answers
Answer:
17
Step-by-step explanation: ..
Domenic and Adriana are observing a hot-air balloon from two tracking stations on the ground.
The tracking stations are 6.0 km apart. From Domenic's point of view, the hot-air balloon is at an
angle of elevation of 72º. From Adriana's point of view, the angle of elevation is 58º.
Answers
Answer:
height of the balloon = 6.32 km
Step-by-step explanation:
The question wants us to find the height of the balloon .
They are observing the balloon from two tracking stations on the ground. The tracking stations are 6 km apart . From Dominic point of view the hair balloon is at an angle of elevation of 72° and from Adriana point of view the elevation is 58° . The illustration forms a triangle containing 2 right angles with a similar height.
To find the height we must find the hypotenuse of one of the right angle triangle. WE are given 2 angle and a side of 6 km. The third angle is 180 - 72 - 58 = 50°.
Using sin rule
6/sin 50° = b/sin 72°
cross multiply
6 sin 72° = b sin 50°
divide both sides by sin 50°
b = 6 sin 72°/sin 50°
b = 6 × 0.95105651629 /0.76604444311
b = 5.70633909777 /0.76604444311
b = 7.44909665372
b ≈ 7.45 km
The height can be found since we know the hypotenuse of one side of one of the right angle triangle.
sin 58° = opposite/hypotenuse
sin 58° = h/7.45
cross multiply
h = 7.45 × sin 58°
h = 7.45 × 0.84804809615
h = 6.31795831637
h ≈ 6.32 km
Who is most affected by a discrepancy in dosage
Answers
When it comes to medication or medical care, the person who would usually face the greatest impact from a discrepancy in dosage is the individual who is undergoing the treatment.
What is the discrepancy in dosageInsufficient dosage may result in a lack of intended therapeutic effects and failure to improve the patient's condition as anticipated. Also, if the amount administered is excessive, the individual may suffer from negative consequences or possible injury.
It should be noted that the effects of a difference in medication dosage can differ based on the type of medicine, the condition being treated, and personal characteristics like age, weight, general health, and specific reactions or sensitivities.
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Which shows the decimal 0.16 expressed as a fraction in the simplest form?
Answers
Answer:
.16 = 16/100= 8/50= 4/25
Step-by-step explanation:
Answer:
decimal: 0.16
fraction: 4/25
percentage: 16%
Step-by-step explanation:
26 Use a letter to represent the unknown, and then write an equation and use it to solve the word problem.
Brooke read 3, 237 pages during the fourth grade.
She read 8, 421 pages in fourth and fifth grades combined.
How many pages did Brooke read in fifth grade?
Part A
Write an equation using the letter p to represent the unknown.
Part B
Use your equation to answer the question.
books.
Answers
Answer: 5,184 pages
Step-by-step explanation:
Equation: 8,421 - 3,237 = p or p + 3,237 = 8,421
Knowing that Brooke read 3,237 pages in the fourth grade, the remainder is going to be the amount of pages she read in the fifth grade.
With the information given and the variable (p) we can use this to create an equation and find the answer.
Hope this helps
[7th Grader]
How can I write this out?
Write the slope intercept form equation of each line.
passes through (7,-2) and (02,)
Answers
y₂-y₁
--------
x₂-x₁.
2 - (-2)
------------
0 - 7
4
---
-7
this is the slope
Imagine a lidless box with height h and a square base whose sides have length x. The box must have a volume of 117 ft^3. Estimate the value of x that produces the box with a minimum surface area.
Answers
The closest value of x that would produce the box with the minimum surface area would be 117 ft.
The volume of a lidless box with height h and a square base with sides of length x is given by V = x^2 * h.
To have a volume of 117 ft^3, we have:
x^2 * h = 117
The surface area of a lidless box is given by:
A = 2x^2 + 4xh
We want to minimize this surface area while still keeping the volume of the box at 117 ft^3. To do this, we need to find an expression for h in terms of x, and then minimize A in terms of x.
From the volume equation, we can solve for h:
h = 117 / x^2
Substituting this expression into the surface area equation, we get:
A = 2x^2 + 4x * (117 / x^2)
= 2x^2 + 466 / x
Now, we can use calculus to find the minimum of A in terms of x.
The derivative of A with respect to x is:
dA/dx = 4x - 466 / x^2
Setting dA/dx equal to zero, we find the critical points:
4x - 466 / x^2 = 0
4x = 466
x = 116.5
We now need to check if x = 116.5 is a local minimum or maximum.
To do this, we need to examine the second derivative of A with respect to x:
d^2A/dx^2 = -932 / x^3
Since the second derivative is negative, this means that A has a local minimum at x = 116.5.
Hence, the value of x that produces the box with a minimum surface area is approximately 116.5 ft.
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Two students drive separate cars.
• Student 1 begins driving 90 minutes before student 2.
• Student 1 drives at a speed of 48 miles per hour.
• Student 2 drives at a speed of 64 miles per hour.
For how many hours has student 1 been driving when both students have driven exactly the same distance?
O 1 hour 30 minutes
3 hours 0 minutes
6 hours 0 minutes
7 hours 30 minutes
Answers
Student take 7hours30 mins to drive.
leshia is cooking the family dinner. The first recipe she is using requies 1/2 a cup of milk and the second requires 4/5 a cup of milk. How much milk does leshia need to cook the dinner?
A 5/7
B 1 3/10
C 1 7/10
D 1 9/10
Answers
Answer:
B.) 1 3/10
Step-by-step explanation:
first, you have to make the denominator the same. Multiply 5 all numbers by two, and then add the numerators. All that’s left is to simplify
write the equation of a line that passes through the point (4,-6) and is parallel to a line that passes through the points(-2, 3.5) and (4,5)
Answers
Answer:
No, the equation does not apply in this case
Step-by-step explanation:
Draw a line through the points (-4,-6) and(4,5), then
a line from any point of the first line to a terminal
point of the second
what’s the answer to 22= b/2 -2
Answers
b would equal 48
_________________
22 + 2 = b/2
24 = b/2
b = 24 x 2 = 48
b = 48
The distribution of monthly charges for cellphone plans in the United States is approximately normal with a mean of $62 and a standard deviation of $18. What percentage of plans have charges that are less than $83.60?
Answers
About 88.49% of cellphone plans have charges that are less than $83.60.
How to determine the percentage of plans have charges that are less than $83.60?To determine the percentage of plans that have charges less than $83.60, we need to find the z-score (z) using the given mean and standard deviation, and then look up the corresponding area under the normal distribution curve.
z = (x – μ) / σ
where x = 83.60, mean, μ = 62 and standard deviation, σ = 18
Thus, the z-score of $83.60 is:
z = (83.60 - 62) / 18 = 1.2
Using a standard normal distribution table, we can find that the area to the left of z = 1.20 is 0.8849 or 88.49% (check image attached).
Therefore, about 88.49% of cellphone plans have charges that are less than $83.60.
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If Jeff has 1 Apple and Jessica has 2 apples how many do they have together
Answers
answer: 3
simply add 1 apple from Jeff and 2 apples from Jessica
(1+3) = ?
Thus, the answer is 3 apples all together
dose anyone know how to solve this?
Answers
Answer:
The right answer is 16m
Step-by-step explanation:
4*4=16
Answer:
16 m
Step-by-step explanation:
→ Remember how sides a square has
4
→ Multiply that by the side length of 4
4 × 4 = 16
Write an equation in slooe intercept form for the line that is parallel to y=-4x-3 and passes through the point (-2, 4).
Answers
Answer:
y=-4x-4
Step-by-step explanation:
m=-4
point(-2,4)
y-4 = -4 (x- (-2) )
y-4 = -4 (x+2)
y-4 = -4x -8
y = -4x -4
Step-by-step explanation:
we are going to use quadratic equation for this particular question . so now we are going to change the signs i.e from negative to positive or from positive to negative(-2 will turns to +2) nd +4 will turns to-4)
therefore, (x+2)(x-4) then we na expand the bracket x(x-4)+2(x-4)
x square -4x+2x-8
therefore the equation is nothing but x square-2x-8
 The difference between the product of four and a number and six
Into algebraic expression
Answers
explain- i got it right
Keenan, Ariel, Denise, and Turner went to a baseball game on a bright sunny day, but they only
brought 1 pair of sunglasses! The friends agreed that each person would wear the sunglasses for
an equal amount of time during the 3-hour game.
How long did each person wear the sunglasses?
Answers
Answer:
there is one pair of sunglasses right, okay so there are 4 friends also
and if the game was 3 hours then.........
then 4 divided by 3 that would be uh,,,mm sorry i don't know im only in the 5th grade
Step-by-step explanation:
???
Find the area of the irregular figure (use 3.14 for pi). Show your work. Round to the nearest tenth.
Enter your answer and show all the steps that you use to solve this problem in the space provided.
Answers
Answer:
A ≈ 13.7 ft²
Step-by-step explanation:
the figure is composed of a triangle ( left ) and a semicircle (right )
the area of a triangle is calculated as
area = \(\frac{1}{2}\) bh ( b is the base and h the perpendicular height )
here b = 3.6 and h = 4.8 , then
area = \(\frac{1}{2}\) × 3.6 × 4.8 = 8.64 ft²
the area of a semicircle is half the area of a circle, that is
area = \(\frac{1}{2}\) πr² ( r is the radius )
here diameter = 3.6 , then r = 3.6 ÷ 2 = 1.8
area = \(\frac{1}{2}\) × 3.14 × 1.8²
= 0.5 × 3.14 × 3.24
≈ 5.1 ft² ( to the nearest tenth )
the total area (A) is then the sum of the 2 parts
A = 8.64 + 5.1
≈ 13.7 ft²
cual es la solucion de este numero entero (+4)+(+12)
Answers
Answer:
16
Step-by-step explanation:
12+1=13
13+1=14
14+1=15
15+1=16
1+1=2
2+1=3
3+1=4
---------
4+6=10
10+6=16
(6+6=12)
7. Order the following integers from least to greatest: 0,5,18,100,22
Answers
Answer:
0, 5, 18, 22, 100
Step-by-step explanation:
Order the integers from least to greatest which means that the smallest number to the biggest number. Since the biggest number is 100 it goes at the end and 0 is the smallest number so it goes at the end. After that, all you have to do is find which numbers are bigger and smaller. So, the numbers should go in this order 0, 5, 18, 22, 100.
Hope This Helps :)
Find x and y.
Give reasons to justify your solution
Answers
Answer:
X=32, Y=148
Step-by-step explanation:
X is 32 degrees because it is parallel to that 32 between f and b.
Y is 148 degrees because x and y together make a straight line and a straight line is 180 degrees therefore, 180-32 = 148.
An ice-cream scoop has a shape of spherical cap. An engineer's goal is to maximize the volume of the scoop given that the surface area of the scoop is 36 cm2. Find the radius of the sphere and the height of the spherical cap of maximum volume. (Hint:
the volume of a spherical cap is \(Vcap=\frac{1}{3} \pi h^2(3R-h)\), where R is the radius of the sphere and h is the height of the cap). Find the maximum volume of ice-cream scoop.
Write your answer in terms of π.
Answers
The radius is 5.316 cm, height is 10.362 cm and Maximum volume is
60.583 cm³.
We have,
Surface area of sphere = 36 cm²
4πr² = 36
πr² = 9
r = 3√π
r= 5.316
Now, Volume spherical cap
= 1/3 πh² (3R- h)
= 1/3π (10.362) (3 (5.316) - 10.362)
= 1/3 π (57.882132)
= 60.583 cm³
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Use the graph above to demonstrate your understanding of slope and density.
a. Calculate the slope of the trend line using this mathematical equation: Slope = Y2-Y1 / X2-X1
(Use the 2 yellow dots on the graph and show your work)
b. In this graph (mass vs volume), what does the slope represent?
Answers
Answer:
41
Step-by-step explanation:
i know skxkdkckxokdkakd
4) the table given bellow shows the number of votes received by the candidantes op school pref2ct in a school candidates A 30 B 80 C 50 D 100
Answers
The ratio of the number of votes received by candidate E to the number of votes received by candidate B is unknown.
What is ratio?A ratio is a mathematical expression that compares two or more related values. Ratios are often expressed as fractions, with the first number representing the numerator and the second number representing the denominator. Ratios are used to compare items, measure relationships, and make comparisons between values. Ratios are often used in business and finance to measure things like profits, debt, and liquidity. They can also be used to compare the size of two items or to measure how much of one item is used in relation to another.
There are four candidates in the school district's election (A, B, C, and D), and the table shows the number of votes received by each candidate. To calculate the ratio of the number of votes received by candidate E to the number of votes received by candidate B, we will use the following formula:
Ratio = (Number of votes for E)/(Number of votes for B)
Since we don't know the number of votes for candidate E, the ratio cannot be calculated. Therefore, the ratio of the number of votes received by candidate E to the number of votes received by candidate B is unknown.
Complete questions as follows-
4) the table given bellow shows the number of votes received by the candidantes op school pref2ct in a school candidates A 30 B 80 C 50 D 100
What is the ratio of number of votes received by E to the number of votes received by B?
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Candidate equation A received 30 votes, Candidate B received 50 votes, Candidate C received 80 votes, and Candidate D received 100 votes.
What is equation?A assertion that two statements are equivalent is referred to as an equation. An equation is made up of two parts separated by the algebraic symbol (=). As an example, the claim 2x+3x=9 states that
"2x + 3 equals the number "9."
The aim of equation solving is to find the value or numbers of the variable(s) required for the equation to be true. Equations can be simple or complex, regular or nonlinear equation, and have one or more components.
The equation \(x^2+2x-3=0\) increases the variable x to the power of two. Lines are used in many various branches of arithmetic including calculus, algebra, and geometry.
The table you supplied appears to be incorrectly structured; however, based on the information you provided, here is what I believe:
The table shows how many ballots each of the four candidates for school prefect (A, B, C, and D) received.
As a result, Candidate A received 30 votes, Candidate B received 50 votes, Candidate C received 80 votes, and Candidate D got 100 votes.
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The complete question is,
The table given bellow shows the number of votes received by the candidantes op school pref2ct in a school
Candidates A B C D
No.of votes 30 80 50 100
Calculate the above information and find each candidate receive how many votes.
Which triangle is a translation of triangle P? On a coordinate plane, triangle P is shifted 5 units to the right and 7 units up to form triangle C. triangle A triangle B triangle C triangle D Mark this and return
Answers
Answer: Triangle C.
Step-by-Step Explanation:
To solve this problem, we need to understand what it means for a triangle to be a translation of another triangle. A translation for a triangle means that it is the same shape and size as the original triangle, but it has shifted (or moved) a specific number of units either up, down, left, or right.
In this problem, we have Triangle P, which has been shifted 5 units to the right and 7 units up to form Triangle C. To determine which triangle is a translation of Triangle P, we can look at the coordinates of the different triangles.
Triangle A: (1, 4), (3, 7), (5, 6)
Triangle B: (2, 3), (4, 6), (6, 5)
Triangle C: (6, 11), (8, 14), (10, 13)
Triangle D: (3, 5), (5, 8), (7, 7)
If we compare the coordinates of each triangle with those of Triangle P, we can see that Triangle C, with coordinates (6, 11), (8, 14) and (10, 13), is the translation of Triangle P because all of the coordinates of Triangle P have been shifted 5 units to the right, and 7 units up. Therefore, the answer is Triangle C.
A submarine is descending at a rate of 25 feet per minute. What is the relationship between the depth of the submarine and the number of minutes that have passed?a. proportionalb.cannot be determinedc. non-proportionald. ratio
Answers
In this case, we have a proportional relationship because the variables represent the same relationship, they're ratios are proportional.
Calculate.
12C4
Note: Cr=
n
n!
r!(n−r)!
Answers
Answer:
495
Step-by-step explanation:
using the definition
n\(C_{r}\) = \(\frac{n!}{r!(n-r)!}\)
where n! = n(n - 1)(n - 2) ... × 3 × 2 × 1
then
12\(C_{4}\)
= \(\frac{12!}{4!(12-4)!}\)
= \(\frac{12!}{4!(8!)}\)
cancel 8! on numerator/ denominator
= \(\frac{12(11)(10)(9)}{4!}\)
= \(\frac{11880}{4(3)(2)(1)}\)
= \(\frac{11880}{24}\)
= 495