Sammy can paint the house by himself in 15 hours. With Nolan's help, it only takes 9 hours.
How long would it take for Nolan to paint the house by himself?

Given:

Sammy can paint the house by himself in 15 hours. With Nolan's help, it only takes 9 hours.

We need to determine the time it would take for Nolan to paint the house by himself.

Time taken by Nolan:

Let the rate that Sammy can paint the house by himself is

The rate that Sammy and Nolan together paint the house is

To determine the rate that it takes for Nolan to paint the house by himself is to subtract the rates.

Thus, we have;

Thus, the rate that it would take Nolan to paint the house by himself is

The time that it takes for Nolan to paint the house by himself can be determined using the formula,

Substituting the values, we have;

Thus, Nolan takes 22.5 hours to paint the house by himself.

step-by-step explanation:

part a. the differences are decreasing, so we know the sequence is not arithmetic. the ratios are constant (=3/4), so we know the sequence is geometric.

part b. from part a, we know the common ratio is 3/4, so each term is 3/4 of the one before. the initial term is the number of gallons at h=1.

g[h] = (3/4)·g[h-1]

g[1] = 384 . . . . initial value required for a recursive definition

part c. the general formula for a geometric sequence is

a[n] = a[1]·r^(n-1)

we have a[1] = 384, r = 3/4, and the term number given by h. the formula that follows is

g[h] = 384·(3/4)^(h-1)

part d. filling in the value and doing the arithmetic, we have

g[12] = 384·(3/4)^11 ≈ 384·0.042235136 ≈ 16.2183 . . gallons

part e. each gallon is 4 quarts, so

16.2183 gal × 4 qt/gal = 64.8732 qt