QuestionA ball is thrown from 7 feet high. Its height is modeled by $f(x)=-0.4 x^{2}+2.7 x+7$ . Find the max height and distance from launch.

Studdy Solution

STEP 1

Assumptions1. The height of the ball is modeled by the function $f(x) = -0.4x^ +.7x +7$ . The variable $x$ represents the ball's horizontal distance in feet from where it was thrown3. We are looking for the maximum height of the ball and the distance from where it was thrown when this occurs

STEP 2

The maximum height of the ball can be found by finding the vertex of the parabola defined by the function $f(x)$ . The x-coordinate of the vertex of a parabola given by the equation $f(x) = ax^2 + bx + c$ is given by $-\frac{b}{2a}$ .

STEP 3

Substitute the values of $a$ and $b$ from the equation $f(x) = -0.x^2 +2.7x +7$ into the formula $-\frac{b}{2a}$ to find the x-coordinate of the vertex.
$x = -\frac{2.7}{2(-0.)}$

STEP 4

Calculate the x-coordinate of the vertex.
$x = -\frac{2.7}{2(-0.4)} =3.375$

STEP 5

The x-coordinate of the vertex represents the horizontal distance from where the ball was thrown when it reaches its maximum height. So, the ball reaches its maximum height3.375 feet from where it was thrown.

STEP 6

To find the maximum height of the ball, substitute the x-coordinate of the vertex into the equation $f(x)$ .
$f(3.375) = -0.4(3.375)^2 +2.(3.375) +$

STEP 7

Calculate the maximum height of the ball.
$f(3.375) = -0.4(3.375)^2 +2.7(3.375) +7 \approx10.6$ So, the maximum height of the ball is approximately10.6 feet, which occurs3.375 feet from the point of release.

Was this helpful?

QuestionA ball is thrown from 7 feet high. Its height is modeled by $f(x)=-0.4 x^{2}+2.7 x+7$ . Find the max height and distance from launch.

Studdy Solution

STEP 1

Assumptions1. The height of the ball is modeled by the function $f(x) = -0.4x^ +.7x +7$ . The variable $x$ represents the ball's horizontal distance in feet from where it was thrown3. We are looking for the maximum height of the ball and the distance from where it was thrown when this occurs

STEP 2

The maximum height of the ball can be found by finding the vertex of the parabola defined by the function $f(x)$ . The x-coordinate of the vertex of a parabola given by the equation $f(x) = ax^2 + bx + c$ is given by $-\frac{b}{2a}$ .

STEP 3

Substitute the values of $a$ and $b$ from the equation $f(x) = -0.x^2 +2.7x +7$ into the formula $-\frac{b}{2a}$ to find the x-coordinate of the vertex.
$x = -\frac{2.7}{2(-0.)}$

STEP 4

Calculate the x-coordinate of the vertex.
$x = -\frac{2.7}{2(-0.4)} =3.375$

STEP 5

The x-coordinate of the vertex represents the horizontal distance from where the ball was thrown when it reaches its maximum height. So, the ball reaches its maximum height3.375 feet from where it was thrown.

STEP 6

To find the maximum height of the ball, substitute the x-coordinate of the vertex into the equation $f(x)$ .
$f(3.375) = -0.4(3.375)^2 +2.(3.375) +$

STEP 7

Calculate the maximum height of the ball.
$f(3.375) = -0.4(3.375)^2 +2.7(3.375) +7 \approx10.6$ So, the maximum height of the ball is approximately10.6 feet, which occurs3.375 feet from the point of release.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionA ball is thrown from 7 feet high. Its height is modeled by f(x)=−0.4x2+2.7x+7f(x)=-0.4 x^{2}+2.7 x+7f(x)=−0.4x2+2.7x+7. Find the max height and distance from launch.

Studdy Solution

STEP 1

STEP 2

The maximum height of the ball can be found by finding the vertex of the parabola defined by the function f(x)f(x)f(x). The x-coordinate of the vertex of a parabola given by the equation f(x)=ax2+bx+cf(x) = ax^2 + bx + cf(x)=ax2+bx+c is given by −b2a-\frac{b}{2a}−2ab​.

STEP 3

Substitute the values of aaa and bbb from the equation f(x)=−0.x2+2.7x+7f(x) = -0.x^2 +2.7x +7f(x)=−0.x2+2.7x+7 into the formula −b2a-\frac{b}{2a}−2ab​ to find the x-coordinate of the vertex.x=−2.72(−0.)x = -\frac{2.7}{2(-0.)}x=−2(−0.)2.7​

STEP 4

Calculate the x-coordinate of the vertex.x=−2.72(−0.4)=3.375x = -\frac{2.7}{2(-0.4)} =3.375x=−2(−0.4)2.7​=3.375

STEP 5

The x-coordinate of the vertex represents the horizontal distance from where the ball was thrown when it reaches its maximum height. So, the ball reaches its maximum height3.375 feet from where it was thrown.

STEP 6

To find the maximum height of the ball, substitute the x-coordinate of the vertex into the equation f(x)f(x)f(x).f(3.375)=−0.4(3.375)2+2.(3.375)+f(3.375) = -0.4(3.375)^2 +2.(3.375) +f(3.375)=−0.4(3.375)2+2.(3.375)+

STEP 7

Calculate the maximum height of the ball.f(3.375)=−0.4(3.375)2+2.7(3.375)+7≈10.6f(3.375) = -0.4(3.375)^2 +2.7(3.375) +7 \approx10.6f(3.375)=−0.4(3.375)2+2.7(3.375)+7≈10.6So, the maximum height of the ball is approximately10.6 feet, which occurs3.375 feet from the point of release.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionA ball is thrown from 7 feet high. Its height is modeled by $f(x)=-0.4 x^{2}+2.7 x+7$ . Find the max height and distance from launch.

The maximum height of the ball can be found by finding the vertex of the parabola defined by the function $f(x)$ . The x-coordinate of the vertex of a parabola given by the equation $f(x) = ax^2 + bx + c$ is given by $-\frac{b}{2a}$ .

Substitute the values of $a$ and $b$ from the equation $f(x) = -0.x^2 +2.7x +7$ into the formula $-\frac{b}{2a}$ to find the x-coordinate of the vertex.
$x = -\frac{2.7}{2(-0.)}$

Calculate the x-coordinate of the vertex.
$x = -\frac{2.7}{2(-0.4)} =3.375$

To find the maximum height of the ball, substitute the x-coordinate of the vertex into the equation $f(x)$ .
$f(3.375) = -0.4(3.375)^2 +2.(3.375) +$

Calculate the maximum height of the ball.
$f(3.375) = -0.4(3.375)^2 +2.7(3.375) +7 \approx10.6$ So, the maximum height of the ball is approximately10.6 feet, which occurs3.375 feet from the point of release.