Question

make sure to show your work.

1. the total rainfall, after t hours of continuous rain, is given by r(t). the depth of the water in a swimming pool, in inches, based upon the amount of rainfall r is given by d(r). interpret the statement d(r(5)) = 4.2 in the context of this problem.
2. the percent of skills mastered after studying for m minutes is given by k(m). the expected score on an exam based upon k percent of skills mastered is given by s(k). interpret the statement s(k(45)) = 92 in the context of this problem.
3. the area of a circle depends on its radius and is given by the function a(r)=\\(\pi r^{2}\\). pepperoni palace charges for pizza by the square - inch according to the function p(a) = 0.25a where a is the area in square inches. write an equation for the function that gives the price of a pizza of radius r inches at pepperoni palace.
4. a table of selected values is given for the functions f and g. the domain of both functions is all real numbers.

x	f(x)
-5	2
-3	4
0	12
1	8
5	4
x	g(x)
0	-1
2	-3
4	5
8	7

find f(g(2)).

1. let f(x)=4 - 2\\(x^{2}\\) and g(x)=x + 3.

a. find f(g(-1)).
b. write an expression for f(g(x)).
c. find g(f(1)).
d. write an expression for g(f(x)).

1. if h(g(x))=\\(\sqrt{2x + 5}\\), find a possible expression for h(x) and g(x).

Explanation:

1.

$R(t)$ gives total rainfall after $t$ hours. $R(5)$ is total rainfall after 5 hours. $D(r)$ gives water - depth in pool based on rainfall $r$. So $D(R(5)) = 4.2$ means that after 5 hours of continuous rain, the depth of the water in the swimming pool is 4.2 inches.

$K(m)$ gives the percent of skills mastered after studying for $m$ minutes. $K(45)$ is the percent of skills mastered after studying for 45 minutes. $S(k)$ gives the expected exam - score based on the percent of skills mastered $k$. So $S(K(45)) = 92$ means that after studying for 45 minutes, the expected score on the exam is 92.

Step1: First find area formula

The area of a circle is $A(r)=\pi r^{2}$, where $r$ is the radius. The price function of pizza is $P(a) = 0.25a$, where $a$ is the area.

Step2: Substitute area formula into price formula

Substitute $a = A(r)=\pi r^{2}$ into $P(a)$. We get $P(A(r))=0.25\pi r^{2}$.

Answer:

After 5 hours of continuous rain, the depth of the water in the swimming pool is 4.2 inches.

2.