Question

Holly plays tennis on the school tennis team. She can hit a tennis ball really high in the air. She learned that the quadratic equation (not the quadratic formula), y=5x^2+40x could be used to determine the height of a tennis ball after it was it with a tennis racket (y=height of the tennis ball in feet, x = seconds after the ball is hit with the racket).

A) How long does it take for the ball to hit the ground?
B) What is the highest point reached during the flight of the ball? how do you know?
C) When was the ball exactly 75 feet off the ground?

Answer 1

The given equation is

`y=-5x^(2) +40x`

Which models the movement of the tennis ball.

(A)

It's important to mention, when the ball hits the ground, the height is zero, that is
`y=0`, and the equation is

`0=-5x^(2) +40x`

We need to solve for
`x`

`5x(8-x)=0`

Using the zero property

`5x=0 \implies x=0\\8-x=0 \implies x=8`

Therefore, the ball hit the ground after 8 seconds.

(B)

The highest point reached by the ball it's the vertex of the parabola of the given equation, which has coordinates of
`V(h,k)`, where

`h=-(b)/(2a)` and
`k=f(h)`

From the equation, we have
`a=-5` and
`b=40`, replacing these values, we have

`h=-(40)/(2(-5))=(40)/(10) =4`

Then, we replace this value to find the vertical coordinate

`k=f(4)=-5(4)^(2) +40(4)=-5(16)+160=80`

Therefore, the highest point reached is 80 feet.

(C)

To find the time when the ball was 75 feet off the ground, we need to substitute and solve.

`y=-5x^(2) +40x\\75=-5x^(2) +40x\\-5x^(2) +40x-75=0\\5x^(2)-40x+75=0`

Let's divide the equation by 5

`x^(2) -8x+15=0`

Now, we need to find two numbers which product is 15 and which sum is 8, those numbers are 5 and 3.

`x^(2) -8x+15=(x-5)(x-3)=0`

Using the zero property, we have

`x-5=0 \implies x=5\\x-3=0 \implies x=3`

Therefore, the ball was 75 feet off the ground at 3 seconds and 5 seconds.

Answer 2

Answers:

A. x = 8 seconds

B. y = 80 feet . We know because it is the peack of the graph.

C. x = 3 s, x=5 s

Explanation:

To easily solve this three part problem, we use a graphing tool to plot the equation that describes the height of the tennis ball over time, after it was hit by a tennis racket. Please, see the attached graph for clarification.

y = -5x^2 + 40*x

(the equation must have the minus sign, otherwise it makes no sense for the problem)

Where,

y = height of the tennis ball in feet,

x = seconds after the ball is hit with the racket

Question A

How long does it take for the ball to hit the ground?

By examining the graph, we need to find the points where the graph intersects with the x-axis.

This is equivalent to making y = 0 in the equation and solving for x

0 = -5x^2 + 40*x

We get two solutions, but time must be positive so, x > 0

The graph shows that when x = 8 seconds, y = 0

-5(8)^2 + 40*(8) = 0

Question B

What is the highest point reached during the flight of the ball? how do you know?

By examining the graph, we need to find the point where the graph has maximum amplitude.

The point corresponds to (4,80), y = 80 feet at x = 4 seconds

We know because it is the maximum peak of the plot. Please see attached picture for reference.

Question C

When was the ball exactly 75 feet off the ground?

We look for y = 75 feet in the plot.

This corresponds to x = 3 seconds and x=5 seconds

This can also be found analytically

75 = -5x^2 + 40*x

75 = -5(3)^2 + 40*(3)