# 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. ?

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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.

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.

##### 1 Answer

**i)**

**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.#