Studdy AI Logo
Math

QuestionFind the maximum height of the ball given the function f(x)=−0.7x2+2.7x+5f(x)=-0.7 x^{2}+2.7 x+5 and its distance from the throw point.

Studdy Solution

STEP 1

Assumptions1. The function f(x)=−0.7x+.7x+5f(x)=-0.7 x^{}+.7 x+5 represents the height of the ball in feet as a function of its horizontal distance from where it was thrown. . We are asked to find the maximum height of the ball and the horizontal distance from the throw point where this maximum height occurs.

STEP 2

The function f(x)=−0.7x2+2.7x+5f(x)=-0.7 x^{2}+2.7 x+5 is a quadratic function, which has the general form f(x)=ax2+bx+cf(x)=ax^{2}+bx+c. The maximum or minimum value of a quadratic function occurs at the vertex of the parabola.

STEP 3

The x-coordinate of the vertex of a parabola given by f(x)=ax2+bx+cf(x)=ax^{2}+bx+c is given by −b2a-\frac{b}{2a}.

STEP 4

Substitute a=−0.7a=-0.7 and b=2.7b=2.7 into the formula to find the x-coordinate of the vertex.
x=−b2a=−2.72(−0.7)x = -\frac{b}{2a} = -\frac{2.7}{2(-0.7)}

STEP 5

Calculate the x-coordinate of the vertex.
x=−2.72(−0.7)≈1.93x = -\frac{2.7}{2(-0.7)} \approx1.93

STEP 6

Now that we have the x-coordinate of the vertex, we can find the y-coordinate, which represents the maximum height of the ball. We do this by substituting the x-coordinate into the function f(x)f(x).
f(1.93)=−0.⋅(1.93)2+2.⋅1.93+5f(1.93) = -0. \cdot (1.93)^{2}+2. \cdot1.93+5

STEP 7

Calculate the y-coordinate of the vertex, which is the maximum height of the ball.
f(1.93)=−0.7⋅(1.93)2+2.7⋅1.93+5≈6.39f(1.93) = -0.7 \cdot (1.93)^{2}+2.7 \cdot1.93+5 \approx6.39The maximum height of the ball is approximately6.39 feet, which occurs approximately1.93 feet from the point of release.

Was this helpful?

Studdy solves anything!

banner

Start learning now

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

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord