Help me on this please

x² = 4py is the parabola's equation, which has an upward-opening focus (0,p) and directrix y = -p. The distance between the directrix and a point on the parabola is set equal to the distance between the focus and the same point on the parabola.

To derive the equation of the parabola using the focus and directrix, we use the definition of a parabola: the set of all points that are equidistant from the focus and the directrix.

Let (x,y) be any point on the parabola. Then, the distance from this point to the focus (0,p) is:

√((x-0)² + (y-p)²) = √(x² + (y-p)²)

The distance from this point to the directrix y=-p is:

|y-(-p)| = |y+p| = y+p

Since the point (x,y) is equidistant from the focus and the directrix, we have:

√(x² + (y-p)²) = y + p

Squaring both sides, we get:

x² + (y-p)² = (y+p)²

Simplifying, we get:

x² = 4py

This is the equation of the parabola with focus (0,p) and directrix y=-p, which opens upwards.

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Complete question

FILL IN THE BLANKS AND SOLVE EQUATION.

On a coordinate plane, a parabola opens upwards. the focus is at point (0, p), point p on the parabola is at (x, y), and point q on the directrix is (x, negative p). the equation of the directrix line is y = - p. when the focus and directrix are used to derive the equation of a parabola, two distances were set equal to each other. = √ (x -x)² (y - (-p))² = √(x -0)² (y - p)²  the distance between the directrix and                 is set equal to the distance between the               and the same point on the parabola. What will be the equation of parabola with focus (0,p) and directrix y=-p, which opens upwards.

Answer:

Step-by-step explanation:

3(2x-4)-5(x+3)/3

6x-12-(5x)/3-1

6x-5x/3-13

13x/3 - 13

13(x/3 - 1) its simplest form of given expression

\(\cfrac{7}{9}(-3^3)+5\implies \cfrac{7}{9}(-1\cdot 3^3)+5\implies \cfrac{7}{9}(-1\cdot 27)+5\implies \cfrac{7}{9}(-27)+5 \\\\\\ \cfrac{7(-27)}{9}+5\implies \cfrac{-189}{9}+5\implies -21+5\implies -16\)

Answer:

-16

Step-by-step explanation:

7/9(-3^3) + 5

7/9(-27) + 5

7/9 x -27 = -21

-21 + 5 = -16

(Hope this helps :) )

The predicted fuel efficiency for a speed of 30 mph will be 26.50 mpg when the least-squares regression line is given.

If the data demonstrates a stronger link between two variables, the line that best matches this linear relationship is known as a least-squares regression line, and it minimizes the vertical distance between the data points and the regression line. A regression line is a straight line that illustrates how a response variable y varies when an explanatory variable x changes. The line is a mathematical model that predicts the value of y given a value of x. A regression line predicts the value of y for a given value of x. A regression line is discovered through regression analysis. When the explanatory variable changes, the regression line shows how much and in which direction the response variable changes.

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Answer:

The equation of the line in the slope-intercept form will be:

Step-by-step explanation:

The slope-intercept form of the line equation

y = mx+b

where  

From the attached graph, taking two points

Finding the slope between (0, 0) and (-10, 70)

\(\mathrm{Slope}=\frac{y_2-y_1}{x_2-x_1}\)

\(\left(x_1,\:y_1\right)=\left(0,\:0\right),\:\left(x_2,\:y_2\right)=\left(-10,\:70\right)\)

\(m=\frac{70-0}{-10-0}\)

\(m=-7\)

Determining the y-intercept:

We know that the value of y-intercept can be determined by setting x = 0, and determining the corresponding value of y.

From the graph, it is clear that:

at x = 0, y = 0

Thus, the y-intercept b = 0

Now, substituting m = -7 and b = 0 in the slope-intercept form

y = mx+b

y = -7x + 0

y = -7x

Therefore, the the equation of the line in slope-intercept form will be:

Answer:

The slope is 3/4 and the y-intercept is -5.

Explanation:

The slope-intercept form of the equation of a line is given as;

where m represents the slope and b represents the y-intercept.

So if we're given the below equation;

To find the slope and y-intercept, all we need to do is to compare the given equation with the standard slope-intercept form of the equation of a line. So if we do that, we'll notice that;

So the slope of the line is 3/4 and the y-intercept is -5.

If the standard deviation of a set of data is 6, then the value of variance is 36

The formula for determining variance is variance = √Standard deviation

Variance of a set of data is equal to square of the standard deviation.

If the standard deviation of a set of data is 6 then we get variance by putting the value of standard deviation in the formula

variance = √Standard deviation

Take square root on both sides

Standard deviation² = 6²

Standard deviation= 36

Hence,  standard deviation of a set of data is 6, then the value of variance is 36

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That means , Seth gain loss of  $261 on his selling stocks.

Last month, Seth made money by selling stocks. His investment income was $401 This month, Seth's investment income is −$261 What does this mean?

Profit(P)

The amount gained by selling a product for more than its cost price.

Loss(L)

The amount the seller incurs after selling the product less than its cost price is mentioned as a loss.

Cost Price (CP)

The amount paid for a product or commodity to purchase is called a cost price. Also, denoted as CP. This cost price is further classified into two different categories:

Fixed Cost: The fixed cost is constant, it doesn’t vary under any circumstances

Variable Cost: It could vary depending on the number of units and other factors

Selling Price (SP)

The amount for which the product is sold is called the Selling Price. It is usually denoted as SP. Also, sometimes called a sale price.

Marked Price Formula (MP)

This is basically labelled by shopkeepers to offer a discount to the customers in such a way that,

Discount = Marked Price – Selling Price

And Discount Percentage = (Discount/Marked price) x 100

Profit and Loss Formulas

Now let us find the profit formula and loss formula.

The profit or gain is equal to the selling price minus the cost price.

Loss is equal to the cost price minus the selling price.

Profit or Gain = Selling price – Cost Price

Loss = Cost Price – Selling Price

The formula for the profit and loss percentage is:

Profit percentage (P%) = (Profit /Cost Price) x 100

Loss percentage (L%) = (Loss / Cost price) x 100

Last month

Seth made money by selling stocks.

His investment income was $401

That means , Seth gain profit of $401 on his selling stocks.

This month

Seth's investment income is −$261

That means , Seth gain loss of  $261 on his selling stocks.

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Answer:

A 100% increase in a total means the final amount is 200% of the initial amount.

A 100% decrease in a total means the final amount is 0% of the initial amount.

Error means that you were a way off from the real value

Step-by-step explanation:

The turning point of the function in this problem is given as follows:

(-5,2).

The function for this problem is defined as follows:

\(f(x) = 2\sqrt[3]{x + 5} + 1\)

The parent function is the cube root function, which has a turning point at (0,0).

The transformations are given as follows:

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The area of the polygon composed of rectangles and triangle is 405.5 ft²

Area is the amount of space occupied by a two dimensional shape or object.

For the first rectangle:

length = 9 ft, width = 5 ft

Area of first rectangle = length * width = 9 ft * 5 ft = 45 ft²

For the second rectangle:

Length = 13 ft, width = (13 + 5) = 18 ft

Area of second rectangle = 13 ft * 18 ft = 234 ft²

For the triangle:

Base = (13 + 9) = 22 ft, height = 29.5 - 13 - 5 = 11.5 ft

Area of triangle = (1/2) * base * height = 0.5 * 22 * 11.5 = 126.5 ft²

Area of polygon = 45 + 234 + 126.5 = 405.5 ft²

The area of the polygon is 405.5 ft²

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a)

It crosses the x-axis 3 times, at x = -3 and x = 1 (multiplicity 2)

Therefore, it has 3 zeros, one of them is repeated

The zeros are:

-3 , 1, 1

b)

Since it has 3 zeroes, the degree is 3

c)

The function tends to infinity as x approaches to infinity

d)

-3 : Multiplicity 1

1 : Multiplicity 2

e)

The y-intercept is the point where the graph crosses the y-axis, as we can see, the point is:

f)

Using the zeros, we can write the equation as:

The vertical component of the final velocity (vfy) of the motorcycle is approximately 11.82 m/s. The magnitude of the final velocity (vf total) of the motorcycle is approximately 31 m/s.

A. To find the vertical component of the final velocity (vfy) of the motorcycle, we need to use the principles of projectile motion. Since the motorcycle is launched from the ramp, we can treat it as a projectile in free fall after leaving the ramp. The vertical component of the final velocity can be calculated using the equation:

vfy = v0y + gt,

where v0y is the vertical component of the initial velocity, g is the acceleration due to gravity (9.8 m/s^2), and t is the time of flight.

To calculate v0y, we can use the fact that the initial velocity (v0) of the motorcycle is 31 m/s, and the angle of the ramp (θ) is 16.5 degrees. The vertical component of the initial velocity is given by:

v0y = v0 * sin(θ).

Substituting the values, we have:

v0y = 31 m/s * sin(16.5 degrees) ≈ 8.36 m/s.

The time of flight can be calculated using the equation:

t = 2 * v0y / g.

Substituting the values, we have:

t = 2 * 8.36 m/s / 9.8 m/s^2 ≈ 1.70 s.

Finally, we can calculate vfy:

vfy = v0y + gt = 8.36 m/s + (9.8 m/s^2) * 1.70 s ≈ 11.82 m/s.

Therefore, the vertical component of the final velocity (vfy) of the motorcycle is approximately 11.82 m/s.

B. The magnitude of the final velocity (vf total) can be calculated using the Pythagorean theorem. Since the horizontal component of the velocity remains constant throughout the motion, vf total is equal to the initial velocity (v0) of the motorcycle, which is 31 m/s.

Therefore, the magnitude of the final velocity (vf total) of the motorcycle is approximately 31 m/s.

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Answer:

42.6cm

Step-by-step explanation:

The sum total of all the sides of the shape is the perimeter of the shape. Hence;

Perimeter = x+3 + 4 + x+9 + 2x+5

Perimeter = 4x + 21

Given

Area of the figure = 75cm^2

Area = (x+3)(2x+5) + (2x+1)(6)

75 = 2x²+5x+6x+15+12x+6

75 =  2x²+23x+21

2x²+23x+21-75 = 0

2x²+23x-54 = 0

x = -23±√23²-4(2)(-75)/4

x = -23±√529+600/4

x = -12±√1129/4

x = -12+33.6/4

x = 21.6/4

x = 5.4

Recall that;

Perimeter = 4x+21

Perimeter = 4(5.4) + 21

Perimeter = 21.6+21

Perimeter = 42.6cm

Hence the perimeter is 42.6cm

Answer:

c. mode

Step-by-step explanation:

.....................

Answer:

x = - 2

Step-by-step explanation:

Reflection across the y-axis:

( x , 6 ) ------> ( - x , 6 )

Translation 2 units right and 4 units down:

( - x , 6 ) -------> ( - x + 2 , 6 - 4 ) = ( - x + 2 , 2 )

Dilation by a factor of 2:

( - x + 2 , 2 ) ------> ( ( - x + 2 ) × 2 , 2 × 2 ) = ( 8 , 4 ) ⇒

( - x + 2 ) × 2 = 8

- 2x + 4 = 8

- 2x = 4

x = - 2

Given:

The point is A(-1,3).

To find:

The coordinates of the final image. If the given point reflected in

(a) x-axis

(b) line y=4.

Solution:

(a)

If a figure reflected across the x-axis, then

\((x,y)\to (x,-y)\)

Using this rule, we get

\(A(-1,3)\to A'(-1,-3)\)

Therefore, the coordinates of the final image are A'(-1,-3).

(b)

If a figure reflected across the line y=4, then

\((x,y)\to (x,-(y-4)+4)\)

\((x,y)\to (x,-y+4+4)\)

\((x,y)\to (x,-y+8)\)

Using this rule, we get

\(A(-1,3)\to A'(-1,-3+8)\)

\(A(-1,3)\to A'(-1,5)\)

Therefore, the coordinates of the final image are A'(-1,5).

The explicit formula for the given sequence is an = 3n + 1. The tenth term is 31.

To find the explicit formula for the sequence, we observe that each term is obtained by adding 3 to the previous term. This indicates that there is a linear relationship between the terms.

Let's denote the nth term as an. We can express the nth term in terms of n as follows:

an = a1 + (n - 1)d,

where a1 is the first term, n is the term number, and d is the common difference between consecutive terms.

In our given sequence, the first term a1 is 4 and the common difference d is 3. Substituting these values into the formula, we have:

an = 4 + (n - 1)3,

an = 4 + 3n - 3,

an = 3n + 1.

Now, to find the tenth term, we substitute n = 10 into the explicit formula:

a10 = 3(10) + 1,

a10 = 30 + 1,

a10 = 31.

The explicit formula for the sequence is an = 3n + 1, and the tenth term is 31.

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Answer:

8 cm

Step-by-step explanation:

To find the length of the rectangle's base, form 2 equations using the given information.

Perimeter of rectangle= 2(height +base)

Let the lengths of the height and base of the rectangle be h and b cm respectively.

Perimeter= 80

2(h +b)= 80

h +b= 80 ÷2

h +b= 40 -----(1)

h= 4b -----(2)

Let's solve by substitution!

Substitute (2) into (1):

4b +b= 40

5b= 40

b= 40 ÷5

b= 8

Thus, the length of its base is 8 cm.

This value represents the present worth below which the probability is 0.5, indicating a negative present worth.

To estimate the probability that the present worth is negative using the NORM.INV function in Excel,

we need to calculate the present worth of the project and then determine the corresponding probability using the normal distribution.

The present worth of the project can be calculated by finding the sum of the present values of the annual receipts over the 10-year period, minus the initial investment. The present value of each annual receipt can be calculated by discounting it back to the present using the minimum attractive rate of return (MARR).

Using the given information, the present value of the initial investment is $10,000. The present value of each annual receipt is calculated by dividing the expected return of $1,800 by \((1+MARR)^t\),

where t is the year. We then sum up these present values for each year.

We can use the NORM.INV function in Excel to estimate the probability of a negative present worth. The function requires the probability value, mean, and standard deviation as inputs.

Since we have a mean and standard deviation for the present worth,

we can calculate the corresponding probability of a negative present worth using NORM.INV.

This value represents the present worth below which the probability is 0.5. By using the NORM.INV function,

we can estimate the probability that the present worth is negative based on the given data and assumptions.

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Answer:

C.  69°

Step-by-step explanation:

The sum of interior angles in a triangle is equal to 180

two of the angles are given to find the third one we subtract their sum from 180

180 - (36+75) = 69 degrees

Answer:

it would be at point (6,3)

Step-by-step explanation:

If you were to reflect it over the x-axis you would get (-6,3)

Let's first start with the given that we have ;

\({:\implies \quad \sf \dfrac{dy}{dx}=xy^{2}}\)

Collect like terms in different sides ;

\({:\implies \quad \sf \dfrac{dy}{y^2}=x\: dx}\)

Integrating both sides will yield

\({:\implies \quad \displaystyle \sf \int y^{-2}dy=\int x^{1}dx}\)

\({:\implies \quad \sf \dfrac{y^{-2+1}}{-2+1}=\dfrac{x^{1+1}}{1+1}+C}\)

\({:\implies \quad \sf \dfrac{-1}{y}=\dfrac{x^2}{2}+C}\)

Where, C is the Arbitrary Constant;

Now, as we are given that x = 1, when y = 1, so putting these values we will obtain C = (-3/2) , putting the values ;

\({:\implies \quad \sf \dfrac{-1}{y}=\dfrac{x^2}{2}-\dfrac32}\)

\({:\implies \quad \sf -\dfrac{1}{y}=\dfrac{x^{2}-3}{2}}\)

\({:\implies \quad \sf -y=\dfrac{2}{x^{2}-3}}\)

\({:\implies \quad \bf \therefore \quad \underline{\underline{y=\dfrac{-2}{x^{2}-3}}}}\)

Option B) is the required answer

Answer:

17

Step-by-step explanation:

IQR = Q3 - Q1

1st Quartile: 13

3rd Quartile: 30

IQR = 30 - 13 = 17

The required solution of the given simultaneous equation is x = -4/27 and y = 56/27.

Simultaneous linear equations are two- or three-variable linear equations that can be solved together to arrive at a common solution.

Here,
Given equation,
y=-14x - - - - - - - -(1)

x+2y=4 - - - - - -- - - -(2)

Put the above value of y in equation 2,
x + 2(-14x) = 4
x - 28x = 4
-27x = 4
x = -4/27

Now,
form expression of y,
y=-14x
y = -14(-4/27)
y = 56/27

Thus, the required solution of the given simultaneous equation is x = -4/27 and y = 56/27.

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Answer:

64

Step-by-step explanation:

23

41

____

 6 4

The probability that you will win the trip to Orlando would be = 1

To calculate the probability that you will win the trip to Orlando the formula for probability should be used and this is given below;

Probability = possible outcome/sample space.

The total small space = 7

For first Letter;

Possible outcome = 1

probability = 1/7. This will be done for the total number of letters used to spell Orlando = 1/7+1/7+1/7+1/7+1/7+1/7+1/7 = 7/7 = 1

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When you multiply a negative number by another negative number, the result is always a positive number.

This rule is based on the properties of multiplication and the concept of negative numbers.

When two negatives are multiplied, the negative signs "cancel out" each other. Mathematically, if you have -a multiplied by -b, where a and b are positive values, the result is given by ab.

For example, if you multiply -2 by -3, the product is 6, which is a positive number. This can be explained by the idea that multiplying two numbers with opposite signs results in a negative product, while multiplying two numbers with the same sign (both negative) produces a positive product.

Therefore, when you multiply a negative by a negative, the outcome is a positive number.

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Fuel consumption determines the number of gallons of fuel needed to travel a certain distance. To travel 2625 miles, the tank must be filled 75 / 15 = 5 times.

To solve the given problem, we need to make use of the concept of fuel consumption. This concept determines the number of gallons of fuel needed to travel a certain distance. Hence, let's first calculate the car's fuel consumption to travel one mile.

Step 1: Fuel consumption to travel one mile The car gives 35 miles per gallon. Hence, fuel consumption to travel one mile = 1 / 35 gallon = 0.0286 gallons

Step 2: Number of times the tank must be filled To travel 2625 miles, the total fuel required = 0.0286 x 2625 = 75 gallons Given, the tank capacity = 15 gallons Hence, the tank must be filled 75 / 15 = 5 times to travel 2625 miles.

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