Question

Duru has an ice-cream stand, which she opens on days that are forecasted to be warmer than 10°C.

When open for business, her daily profit, P, is a function of the number of cones she sells, n, and can be
modeled by the function P(n) = 2n - 20. Additionally, she finds that the number of cones she sells, C.
is a function of the average temperature, t, in degrees Celsius (°C), and can be modeled by the function
C(t) = 50 + 10(t – 10).
Find an explicit expression that gives Duru's profit on a day with an average temperature of t° C.

Answer 1

`P(c(t)) = 20t-120`

Explanation:

Given

`P(n) = 2n - 20`

`C(t) = 50 + 10(t - 10)`

Required

Determine
`P(C(t))`

The implication of the question is to solve for
`P(C(t))`

Given that

`C(t) = 50 + 10(t - 10)` and
`P(n) = 2n - 20`

`P(c(t))` would be

`P(c(t)) = 2(50 + 10(t - 10)) - 20`

`P(c(t)) = 2(50 + 10t - 100) - 20`

Collect Like Terms

`P(c(t)) = 2(50 - 100+ 10t) - 20`

`P(c(t)) = 2(-50+ 10t) - 20`

Open Bracket

`P(c(t)) = -100+ 20t - 20`

Collect Like Terms

`P(c(t)) = 20t-100 - 20`

`P(c(t)) = 20t-120`