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QuestionAn athlete releases a shot at 50 degrees, modeled by f(x)=−0.2x2+1.2x+5.3f(x)=-0.2x^2+1.2x+5.3. Find max height and distance.

Studdy Solution

STEP 1

Assumptions1. The path of the shot can be modeled by the function f(x)=−0.x+1.x+5.3f(x)=-0.x^+1.x+5.3 . The maximum height of the shot is23.3 feet3. This maximum height occurs30 feet from the point of release

STEP 2

For part a, we are asked to find the maximum height of the shot and how far from its point of release this occurs. This information has already been provided in the problem.The maximum height of the shot is23. feet and this occurs30 feet from the point of release.

STEP 3

For part b, we need to find the shot's maximum horizontal distance, or the distance of the throw.The maximum horizontal distance occurs when the shot hits the ground, which is when f(x)=0f(x) =0.
So, we need to solve the equation −0.2x2+1.2x+5.3=0-0.2x^2+1.2x+5.3 =0 for xx.

STEP 4

This is a quadratic equation in the form ax2+bx+c=0ax^2+bx+c =0. We can solve it using the quadratic formula x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2-4ac}}{2a}.
In this case, a=−0.2a = -0.2, b=1.2b =1.2, and c=.3c =.3.

STEP 5

Plug in the values for aa, bb, and cc into the quadratic formula to solve for xx.
x=−1.2±(1.2)2−4(−0.2)(5.3)2(−0.2)x = \frac{-1.2 \pm \sqrt{(1.2)^2-4(-0.2)(5.3)}}{2(-0.2)}

STEP 6

implify the expression under the square root.
x=−1.2±1.44+4.24−0.4x = \frac{-1.2 \pm \sqrt{1.44+4.24}}{-0.4}

STEP 7

Calculate the value under the square root.
x=−1.2±5.68−0.4x = \frac{-1.2 \pm \sqrt{5.68}}{-0.4}

STEP 8

Calculate the two possible values for xx.
x=−1.2+5.68−0.4orx=−1.2−5.68−0.4x = \frac{-1.2 + \sqrt{5.68}}{-0.4} \quad \text{or} \quad x = \frac{-1.2 - \sqrt{5.68}}{-0.4}

STEP 9

Calculate the two possible values for xx.
x=−.2+5.68−.4=33.5orx=−.2−5.68−.4=−7.5x = \frac{-.2 + \sqrt{5.68}}{-.4} =33.5 \quad \text{or} \quad x = \frac{-.2 - \sqrt{5.68}}{-.4} = -7.5

STEP 10

Since distance cannot be negative, we discard the negative solution.So, the maximum horizontal distance, or the distance of the throw, is33.5 feet.
To the nearest tenth of a foot, this is33.5 feet.

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