Question

A parabolic arch sculpture is on top of a city bank. A model of the arch is y = −0.005x2 + 0.3x where x and y are in feet.

Answer 1

Answer:

A. a. 34.5 feet

b. 60 feet

Step-by-step explanation:

Parabola equation

$y=-0.005x^2+0.3x$

Differentiating with respect to x we get

$\dfrac{dy}{dx}=-0.01x+0.3$

Equating with zero

$-0.01x+0.3=0\\\Rightarrow -0.01x=-0.3\\\Rightarrow x=\dfrac{0.3}{0.01}\\\Rightarrow x=30$

Double derivative of the parabolic equation

$\dfrac{d^2y}{dx^2}=-0.01$

So, $x=30$ is maximum.

$y=-0.005\times 30^2+0.3\times 30\\\Rightarrow y=4.5$

So, the maximum height of the arch will be 4.5 feet.

From the ground the highest point of the arch will be $30+4.5=34.5\ \text{ft}$

We are taking the x axis as the width of the bank.

$0=-0.005x^2+0.3x\\\Rightarrow 0.005x^2=0.3x\\\Rightarrow x=\dfrac{0.3}{0.005}\\\Rightarrow x=60$

So, the width of the bank will be 60 feet.