Question

this question has two parts. first, answer part a. then, answer part b. part a: the costume - department of a theatre company is making cone - shaped hats for a play set in medieval times. each hat will be covered with satin over its entire lateral surface area inside and out. the slant height of each hat will remain constant at 20 inches, but the radius of the base will vary. make connections: write a function y for the lateral area of the hat as a function of the radius of its base, x. (hint: the formula for the lateral area of a cone is y = πxs, where s is the slant height.) part b: make connections: enter the function into your graphing calculator. press window and enter the following settings: xmin: - 10; xmax: 10; ymin: - 1000; ymax: 1000. then press graph. compare the graph on the calculator to the graph of the function described below. intercept: no lateral area when the radius of its base is 0 inches. nonlinear: the graph of the function is linear for radius of a base greater than 0. positive: for radius of a base greater than 0. increasing: for radius of a base greater than 0. end behavior: as the radius of the base increases, the lateral area increases and have an x - and y - intercept at select choice select choice select choice

Explanation:

Step1: Recall lateral - area formula

The formula for the lateral area of a cone is $y=\pi rs$, where $s = 20$ (constant slant - height). Substituting $s = 20$ into the formula, we get $y = 20\pi x$ (let $x$ be the radius of the base).

Step2: Analyze function properties

This is a linear function of the form $y=mx + b$, where $m = 20\pi$ and $b = 0$.

Step3: Find intercepts

For the $x$ - intercept, set $y = 0$. Then $0=20\pi x$, which gives $x = 0$. For the $y$ - intercept, set $x = 0$. Then $y=0$.

Step4: Analyze end - behavior

Since the coefficient of $x$ ($20\pi\approx20\times3.14 = 62.8>0$) is positive, as $x$ (radius) increases, $y$ (lateral area) increases.

Answer:

The function for the lateral area of the hat is $y = 20\pi x$. It is a linear function with no $y$ - intercept and no $x$ - intercept other than at the origin $(0,0)$. It is positive for $x>0$ and as the radius of the base increases, the lateral area increases.