An average of 6000 passengers a day takes the city subway trains at the cost of $3 per ride.

Question

The city board is thinking about raising the price to $3.50 per ride in order to generate larger revenue. They talk to a consulting team whose study shows that for each $0.50 increase in price, the number of passengers would be reduced by an average of 1000 a day. The consulting team suggests the city board should leave the price of $3 per ride since it already yields the maximum revenue. Prove that the consulting team is correct.

Answer 1

An average of 6000 passengers a day takes the city subway trains at the cost of $3 per ride. The city board is thinking about raising the price to $3.50 per ride in order to generate larger revenue. They talk to a consulting team whose study shows that for each $0.50 increase in price, the number of passengers would be reduced by an average of 1000 a day. The consulting team suggests the city board should leave the price of $3 per ride since it already yields the maximum revenue. Prove that the consulting team is correct.

Detailed Solution:

Cost Passengers Revenue

$3 + (0)$0.50 6000 – (0)(1000) 18000

$3 + (1)$0.50 6000 – (1)(1000) 17500

$3 + (2)$0.50 6000 – (2)(1000) 16000

...

$3 + (x)$0.50 6000 – (x)(1000) (3 + 0.5x)(6000 – 1000x) = R(x)

Revenue: R(x) = (3 + 0.5x)(6000 – 1000x)

To find x that gives you Maximum Revenue:

a) For College Algebra/Algebra 2 students: Convert this R(x) into vertex form to find (h, k): x = h gives you the Maximum Revenue; y = k is the Maximum Revenue. R(x) = a(x – h)² + k==>(h, k) = (0, 18000)

b) For Calculus students - Let's do it together

Take the derivative of R(x) which is R ^' (x), set it equal to zeros, and solve for x. The solution x gives you the Maximum Revenue

R(x) = (3 + 0.5x)(6000 – 1000x)

R ^' (x) = (3 + 0.5x)(-1000) + (6000 – 1000x)(0.5) ==> R ^' (x) = (-3000 – 500x) + (3000 – 500x) ==>

R ^' (x) = (-3000 – 500x) + (3000 – 500x) ==> R ^' (x) = -1000x

For Maximum Revenue: ==> R ^' (x) = 0 ==> -1000x = 0 ==> x = 0 <== means no increase gives Maximum Revenue.

For Maximum Revenue: set the Cost = $3 + (x)$0.50 to Cost = $3 + (0)($0.50) = $3 (no increase)

Cost Passengers Maximum Revenue

$3 + (0)$0.50 6000 – (0)(1000) 18000

$3 6000 $18000

This proves that the consulting team is correct. The consulting team suggests the city board should leave the price of $3 per ride since it already yields the maximum revenue.

Answer 2

x represents each increase

R(x) = (6000 - 1000x)(3 + 3.5x)

Expand, convert to vertex form.