Answer to Question #193579 in Calculus for May

Question

You are building the Great Pyramids of PCA, and you need to know how much stone you need to lay out. You determine that the 5th layer of the pyramid requires 720 ft of stone, and the 21st layer requires 600 ft of stone. Each successive layer of the pyramid differs by a constant length. How many feet of stone you need to build the first 15 layers?

Answer 1

This is an Arithmetic progression problem . Let the constant value be d which is our common difference

5th layer of the pyramid requires 720 ft of stone

$\implies$ $5th = 720=a+(5-1)d=a+4d.......(i)$

21st layer requires 600 ft of stone

$\implies 21st=600=a+(21-1)d=a+20d.......(ii)$

Solve the two equations simultaneously

Subtracting equation (ii) from (i) we have

$-16d=120 \implies d={-120\over 16} = -7.5$

$\implies a=720-4(-7.5)=750$

Where a is the first layer of the pyramid

We are asked to find the number of feets needed for 15 layers

Total feets for 15 layers =${n[2a+(n-1)d]\over 2}$

$={15[(2×750)+(15-1)×(-7.5)]\over 2}=10,462.5 feets$