Jack is a window cleaner. He carries 3 ladders. The shortest ladder is half as long as the middle ladder. The longest ladder is twice as long as the middle ladder. If he joins all of the ladders together he can climb 42 feet. How long is each ladder.

The answer to your question is shortest ladder = 6 ft, middle ladder = 12 ft, longest ladder = 24 ft.

Step-by-step explanation:

Data

Ladder 1 = x/2

Ladder 2 = x

Ladder 3 = 2x

Total length = 42 ft

Process

1.- Sum the length of all ladders and solve for x

                   x/2 + x + 2x = 42

Least common factor = 2

                   (x + 2x + 4x)/2 = 42

Simplify

                   7x/2 = 42

                       7x = 42(2)

                       7x = 84

                         x = 84/7

                         x = 12

2.- Find the length of the ladders

Ladder 1 = 12/2 = 6

Ladder 2 = 12

Ladder 3 = 24                    

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