Atriangular banner has an area 1500 square feet. find the measures of the base and height of the triangle if the base is five-sixths of the height.

Step-by-step explanation:

The formula for the area of a triangle is ...

  A = (1/2)bh

Filling in the numbers you are given, you get ...

  1500 = (1/2)(5/6h)(h)

  3600 = h^2 . . . . . . . . . multiply by 12/5

  60 = h . . . . . . . . . . . . . .square root

  b = 5/6·60 = 50 . . . . . . base is 5/6 of height

The measures of the base and height are 50 ft and 60 ft, respectively.

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