The maximum height of the horseshoe will appear on the vertex and the vertex lies on the axis of symmetry. Therefore, we will first find the axis of symmetry using the formula

x = - \frac{b}{2 a} .

y = - 0.01 x^{2} + 0.3 x + 2

In this case,

a = - 0.01

and

b = 0.3 .

Let's substitute them in the formula and find the axis of symmetry.

x = - \frac{b}{2 a}

SubstituteII

$a = - 0.01$ , $b = 0.3$

x = - \frac{0 . 3}{2 ( - 0 . 0 1 )}

MultPosNeg

$a (- b) = - a \cdot b$

x = - \frac{0 . 3}{- 0 . 0 2}

RemoveNegFracAndDenom

$- \frac{a}{- b} = \frac{a}{b}$

x = \frac{0 . 3}{0 . 0 2}

CalcQuot

Calculate quotient

x = 15

Now that we know the axis of symmetry, we can substitute

x = 15

in the equation and find the maximum height.

y = - 0.01 x^{2} + 0.3 x + 2

Substitute

$x = 15$

y = - 0.01 (15)^{2} + 0.3 (15) + 2

CalcPow

Calculate power

y = - 0.01 (225) + 0.3 (15) + 2

Multiply

y = - 2.25 + 4.5 + 2

AddTerms

Add terms

y = 4.25

Therefore, the maximum height is

4.25

feet. We can find the distance traveled by finding the zeros of the function. To do that we will first substitute

y = 0

and write the resulting equation in standard form.

y = - 0.01 x^{2} + 0.3 x + 2

Substitute

$y = 0$

0 = - 0.01 x^{2} + 0.3 x + 2

RearrangeEqn

Rearrange equation

- 0.01 x^{2} + 0.3 x + 2 = 0

DivEqn

$LHS / (- 1) = RHS / (- 1)$

0.01 x^{2} - 0.3 x - 2 = 0

Now that the equation is in standard form, we can solve it by using the Quadratic Formula.

0.01 x^{2} - 0.3 x - 2 = 0

▼

Solve using the quadratic formula

UseQuadForm

Use the Quadratic Formula: $a = 0.01, b = - 0.3, c = - 2$

x = \frac{- ( - 0 . 3 ) \pm ( - 0 . 3 ) ^{2} - 4 ( 0 . 0 1 ) ( - 2 )}{2 ( 0 . 0 1 )}

MultPosNeg

$a (- b) = - a \cdot b$

x = \frac{- ( - 0 . 3 ) \pm ( - 0 . 3 ) ^{2} - ( - 0 . 0 8 )}{2 ( 0 . 0 1 )}

NegNeg

$- (- a) = a$

x = \frac{0 . 3 \pm ( - 0 . 3 ) ^{2} + 0 . 0 8}{2 ( 0 . 0 1 )}

Multiply

x = \frac{0 . 3 \pm ( - 0 . 3 ) ^{2} + 0 . 0 8}{0 . 0 2}

CalcPow

Calculate power

x = \frac{0 . 3 \pm 0 . 0 9 + 0 . 0 8}{0 . 0 2}

AddTerms

Add terms

x = \frac{0 . 3 \pm 0 . 1 7}{0 . 0 2}

CalcRoot

Calculate root

x = \frac{0 . 3 \pm 0 . 4 1 2 3 1 \dots}{0 . 0 2}

We can simplify this result into two separate roots.

$x = \frac{0 . 3 \pm 0 . 4 1 2 3 1 \dots}{0 . 0 2}$
$x_{1} = \frac{0 . 3 \pm 0 . 4 1 2 3 1 \dots}{0 . 0 2}$	$x_{2} = \frac{0 . 3 \pm 0 . 4 1 2 3 1 \dots}{0 . 0 2}$
$x_{1} \approx - 5.62$	$x_{2} \approx 35.62$

Because $x$ is a measurement, it cannot be negative. Therefore, the distance traveled is $35.62$ feet.

Exercise 14 Page 151

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