QuestionA golf ball is hit at 130 ft/s at $45^{\circ}$ . Find the distance $x$ it travels using $h(x)=\frac{-32 x^{2}}{130^{2}}+x$ .

Studdy Solution

STEP 1

Assumptions1. The initial velocity of the golf ball is130 feet per second. . The inclination of the hit is $45^{\circ}$ to the horizontal.
3. The height $h$ of the golf ball is given by the function $h(x)=\frac{-32 x^{}}{130^{}}+x$ 4. $x$ is the horizontal distance that the golf ball has traveled.

STEP 2

To find how far the golf ball was hit, we need to find the value of $x$ when the height $h(x)$ is zero. This is because the golf ball hits the ground when its height is zero. So, we set the function $h(x)$ equal to zero and solve for $x$ .
$0 = \frac{-32 x^{2}}{130^{2}}+x$

STEP 3

To simplify the equation, we can multiply both sides by $130^2$ to get rid of the fraction.
$0 = -32 x^{2} +130^2x$

STEP 4

Rearrange the equation to the standard quadratic form $ax^2 + bx + c =0$ .
$32 x^{2} -130^2x =0$

STEP 5

This is a quadratic equation in the form $ax^2 + bx + c =0$ . We can factor out $x$ to simplify the equation.
$x(32x -130^2) =0$

STEP 6

Set each factor equal to zero and solve for $x$ .
$x =0, \quad32x -130^2 =0$

STEP 7

olving the second equation $32x -130^2 =0$ for $x$ gives us the distance the golf ball was hit.
$x = \frac{130^2}{32}$

STEP 8

Calculate the value of $x$ .
$x = \frac{130^2}{32} \approx526.5625$ The golf ball was hit approximately526.56 feet.

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QuestionA golf ball is hit at 130 ft/s at $45^{\circ}$ . Find the distance $x$ it travels using $h(x)=\frac{-32 x^{2}}{130^{2}}+x$ .

Studdy Solution

STEP 1

Assumptions1. The initial velocity of the golf ball is130 feet per second. . The inclination of the hit is $45^{\circ}$ to the horizontal.
3. The height $h$ of the golf ball is given by the function $h(x)=\frac{-32 x^{}}{130^{}}+x$ 4. $x$ is the horizontal distance that the golf ball has traveled.

STEP 2

To find how far the golf ball was hit, we need to find the value of $x$ when the height $h(x)$ is zero. This is because the golf ball hits the ground when its height is zero. So, we set the function $h(x)$ equal to zero and solve for $x$ .
$0 = \frac{-32 x^{2}}{130^{2}}+x$

STEP 3

To simplify the equation, we can multiply both sides by $130^2$ to get rid of the fraction.
$0 = -32 x^{2} +130^2x$

STEP 4

Rearrange the equation to the standard quadratic form $ax^2 + bx + c =0$ .
$32 x^{2} -130^2x =0$

STEP 5

This is a quadratic equation in the form $ax^2 + bx + c =0$ . We can factor out $x$ to simplify the equation.
$x(32x -130^2) =0$

STEP 6

Set each factor equal to zero and solve for $x$ .
$x =0, \quad32x -130^2 =0$

STEP 7

olving the second equation $32x -130^2 =0$ for $x$ gives us the distance the golf ball was hit.
$x = \frac{130^2}{32}$

STEP 8

Calculate the value of $x$ .
$x = \frac{130^2}{32} \approx526.5625$ The golf ball was hit approximately526.56 feet.

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QuestionA golf ball is hit at 130 ft/s at 45∘45^{\circ}45∘. Find the distance xxx it travels using h(x)=−32x21302+xh(x)=\frac{-32 x^{2}}{130^{2}}+xh(x)=1302−32x2​+x.

Studdy Solution

STEP 1

STEP 2

STEP 3

To simplify the equation, we can multiply both sides by 1302130^21302 to get rid of the fraction.0=−32x2+1302x0 = -32 x^{2} +130^2x0=−32x2+1302x

STEP 4

Rearrange the equation to the standard quadratic form ax2+bx+c=0ax^2 + bx + c =0ax2+bx+c=0.32x2−1302x=032 x^{2} -130^2x =032x2−1302x=0

STEP 5

This is a quadratic equation in the form ax2+bx+c=0ax^2 + bx + c =0ax2+bx+c=0. We can factor out xxx to simplify the equation.x(32x−1302)=0x(32x -130^2) =0x(32x−1302)=0

STEP 6

Set each factor equal to zero and solve for xxx.x=0,32x−1302=0x =0, \quad32x -130^2 =0x=0,32x−1302=0

STEP 7

olving the second equation 32x−1302=032x -130^2 =032x−1302=0 for xxx gives us the distance the golf ball was hit.x=130232x = \frac{130^2}{32}x=321302​

STEP 8

Calculate the value of xxx.x=130232≈526.5625x = \frac{130^2}{32} \approx526.5625x=321302​≈526.5625The golf ball was hit approximately526.56 feet.

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QuestionA golf ball is hit at 130 ft/s at $45^{\circ}$ . Find the distance $x$ it travels using $h(x)=\frac{-32 x^{2}}{130^{2}}+x$ .

To simplify the equation, we can multiply both sides by $130^2$ to get rid of the fraction.
$0 = -32 x^{2} +130^2x$

Rearrange the equation to the standard quadratic form $ax^2 + bx + c =0$ .
$32 x^{2} -130^2x =0$

This is a quadratic equation in the form $ax^2 + bx + c =0$ . We can factor out $x$ to simplify the equation.
$x(32x -130^2) =0$

Set each factor equal to zero and solve for $x$ .
$x =0, \quad32x -130^2 =0$

olving the second equation $32x -130^2 =0$ for $x$ gives us the distance the golf ball was hit.
$x = \frac{130^2}{32}$

Calculate the value of $x$ .
$x = \frac{130^2}{32} \approx526.5625$ The golf ball was hit approximately526.56 feet.