Question

2 companies, A and B, drill wells in a rural area. Company A charges a flat fee of RM3500 to drill a well regardless of its depth. Company B charges RM1000 plus RM12 per foot to drill a well. ?

The depths of wells in this area have a normal distribution with a mean of 250 feet and a standard deviation of 40 feet.

i) What is the probability that Company B would charge more than Company A to drill a well?
ii) Find the mean amount charged by Company B to drill a well.

Answer 1

i) `"P"(X > 208.33)=0.8508, or 85.08%.`
ii) Company B charges an average amount of RM4000.

Explanation:

Part i) First, find out the well depth needed for Company B to charge more than Company A.

This is determined by finding the depth where their costs are equal:

`"A's cost " = " B's cost"`
`"       "3500 = 1000 + 12x" "` (where `x` is number of feet)
`"       "2500 = 12x`

`x=2500/12="RM"208.33`

Since the price for Company A remains fixed, then for any depth exceeding 208.33 feet, Company B charges more than Company A.

Then, find the probability that a well depth exceeds this threshold.

What is the probability that a well depth is greater than 208.33 feet?

Let `X` be the depth of a random well. Then `X" ~ Normal"(mu = 250, sigma^2 = 40^2).`

We seek `"P"(X>208.33)`. This can be translated into a probability for the standard normal variable `Z` using the formula `z=(x-mu)/sigma.`

`z=(x-mu)/sigma = (208.33-250)/40=–1.04175`

Thus,

`"P"(X>208.33)="P"(Z>–1.04175)`
`color(white)("P"(X>208.33))=1-"P"(Z<=–1.04175)`

By table lookup, `"P"(Z <= –1.04) = 0.1492`, so we get

`"P"(X>208.33)=1-0.1492`
`color(white)("P"(X>208.33))=0.8508`.

Part ii) If well depth has an average value of 250 feet, then the mean amount Company B will charge to drill a well is

`1000+12x=1000+12(250)`
`color(white)(1000+12x)=1000+3000`
`color(white)(1000+12x)="RM"4000.`