Question

an automatic soap dispenser holds 18 ounces (oz) of soap. each time someone places their hand under the soap dispenser releases 0.15 oz of soap. the function f(x)=18 - 0.15x models the amount of soap, in ounces, in the dispenser for x dispenses.

1. find the value of x for which f(x)=13.2.
2. in terms of the context, what does the value of x that is the solution to f(x)=13.2 mean?
3. find the x - intercept.
4. what does the x - intercept mean within the context?

Explanation:

Step1: Substitute $f(x)$ value

Set $18 - 0.15x=13.2$ since $f(x) = 13.2$.

Step2: Isolate the term with $x$

Subtract 18 from both sides: $- 0.15x=13.2 - 18=-4.8$.

Step3: Solve for $x$

Divide both sides by - 0.15: $x=\frac{-4.8}{-0.15}=32$.

Answer:

1. $x = 32$
2. The value of $x = 32$ means that after 32 soap - dispenses, there are 13.2 ounces of soap left in the dispenser.
3. To find the $x$-intercept, set $f(x)=0$. So, $18 - 0.15x = 0$. Subtract 18 from both sides: $-0.15x=-18$. Then divide both sides by - 0.15, we get $x = 120$.
4. The $x$-intercept of $x = 120$ means that after 120 soap - dispenses, there is 0 ounce of soap left in the dispenser, i.e., the dispenser is empty.