Question

questions 4 and 5 refer to the following.
a liquid storage tank is in the shape of a right circular cylinder with a radius of 10 feet, as shown. the tank initially contains some water. at time ( t = 0 ), more water starts flowing into the tank. the height of the water in the tank ( t ) seconds after the water starts flowing into the tank is modeled by the function ( h(t)=8 + 0.1 t ), where ( 0 leq t leq 210 ).
suppose that the rate at which the water flows into the tank is tripled and the initial height of the water stays the same. which of the following transformations of function ( h ) correctly expresses a function with the tripled rate and same initial height?
(a) ( \frac{1}{3} h(t) ), because multiplying the output by ( \frac{1}{3} ) has the effect of scaling the ( y )-intercept of function ( h ) by 3.
(b) ( hleft(\frac{1}{3} t
ight) ), because multiplying the input by ( \frac{1}{3} ) has the effect of scaling the constant rate of change of function ( h ) by 3.
(c) ( 3 h(t) ), because multiplying the output by 3 has the effect of scaling ( h ).
(d) ( h(3 t) ), because multiplying the input by 3 has the effect of scaling the constant rate of change of function ( h ) by 3.

Explanation:

Step1: Recall the rules of function transformations

When we have a function \(y = f(x)\) and we consider \(y = af(x)\), if \(a> 1\), it is a vertical stretch and if \(0 < a<1\), it is a vertical compression. The \(y -\)intercept of \(y = f(x)\) is \(f(0)\) and for \(y=af(x)\) it is \(af(0)\). The rate of change of a linear function \(y = mx + b\) is \(m\). If \(y = af(x)\) where \(f(x)=mx + b\), then \(y=a(mx + b)=amx+ab\). The rate of change becomes \(am\) and the \(y -\)intercept becomes \(ab\).

Step2: Analyze the function \(h(t)=0.1t + 8\)

The original function \(h(t)\) has a rate of change (slope) \(m = 0.1\) and \(y -\)intercept \(b = 8\). If we consider a new function \(y = 3h(t)\), then \(y=3(0.1t + 8)=0.3t+24\). The rate of change of \(y = 3h(t)\) is \(0.3\) (which is \(3\times0.1\)) and the \(y -\)intercept is \(24\) (which is \(3\times8\)).

Answer:

C. \(3h(t)\), because multiplying the output by \(3\) has the effect of scaling it.