33.use formulas to find the lateral area and surface area of the given prism round your answer to the nearest whole number.

( answer asap if you have the answers to the rest of the exam i'll take them too )

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Step-by-step explanation:

Part 1) Find the lateral area of the given prism

we know that

The lateral area of the prism is equal to

where

P is the perimeter of the triangular base

L is the length of the prism

Find the perimeter P

we have    

substitute

Part 2) Find the surface area of the given prism

we know that

The surface area of the prism is equal to

where

B is the area of the triangular base

LA is the lateral area of the prism

Find the area B

we have

substitute

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