a bus company has 4000 passengers daily, each paying a fare of $2. for each $0.15 increase, the company estimates that it will lose 40 passengers per day. if the company needs to take in 10 450$ per day to stay in business, what fare should be charged?
increase fare by n 15 cent units
new fare = (2.00 + .15 n)
new passengers = (4000 - 40 n)
so
10450 = (2.00+.15n)(4000-40n)
10450 = 8000 + 520 n - 6 n^2
so
6 n^2 - 520 n + 2450 = 0
3 n^2 - 260 n + 1225 = 0
n = [260 +/- sqrt(67600-14700)]/6
n = [260 +/- sqrt(52900)]/6
n = [260 +/- 230]/6
n = [30]/6 or [490]/6
n = 5 or 81 2/3
well
2.00 + .15(5) = $2.75
which is a reasonable fare
however
2.00 +.15(81 2/3) = 14.25
and 4000 - 40 (81 2/3) = 733 passengers
Well, I guess not many people would ride for $14.25 per day so we better stick with the 75 cent fare increase
To determine the fare that should be charged, we need to find the fare amount that will generate a revenue of $10,450 per day, while taking into account the estimated loss of passengers with each fare increase.
Let's break down the problem step-by-step:
1. Determine the number of passengers for different fare levels:
- For the initial fare of $2, there are 4,000 passengers.
- For each $0.15 increase in fare, the company estimates a loss of 40 passengers.
We can represent the number of passengers (P) in terms of the fare (F) as:
P = 4000 - 40 * ((F - 2) / 0.15)
2. Calculate the revenue generated for different fare levels:
Revenue (R) = Fare (F) * Number of Passengers (P)
3. Set up an equation to find the fare that yields $10,450 in revenue:
R = $10,450
Substitute the expression for revenue and the equation for the number of passengers into the equation:
(F) * (4000 - 40 * ((F - 2) / 0.15)) = $10,450
Now let's solve this equation step-by-step to find the fare amount.
(Note: To simplify the calculations, we can convert all the currency units into cents.)
4. Convert into cents:
Revenue = $10,450 * 100 = 1,045,000 cents
5. Simplify the equation:
(F) * (4000 - 40 * (20F - 40)) = 1,045,000
6. Expand and simplify further:
4000F - 800F^2 + 1600F = 1,045,000
7. Rearrange the equation:
800F^2 - 1600F + 1,045,000 = 0
8. Divide the entire equation by 200 to simplify calculations and make it easier to factor:
4F^2 - 8F + 5,225 = 0
9. Factorize the equation:
(2F - 65)(2F - 65) = 0
10. Solve for F:
2F - 65 = 0
2F = 65
F = 32.50
11. Convert the fare amount back into dollars:
Fare = $32.50 / 100 = $0.3250
Therefore, the fare that should be charged is approximately $0.3250, or 32.5 cents.
To determine the fare that should be charged, we can follow these steps:
1. Calculate the initial daily revenue: We know that there are 4000 passengers daily, and each passenger pays a fare of $2. So, the initial daily revenue can be calculated as follows:
Initial daily revenue = Number of passengers x Fare per passenger
Initial daily revenue = 4000 passengers x $2
Initial daily revenue = $8000
2. Determine the change in daily revenue for a $0.15 increase in fare: According to the information given, for each $0.15 increase in fare, the company expects to lose 40 passengers per day.
3. Calculate the change in daily revenue:
Change in daily revenue = Change in fare x Number of passengers lost per $0.15 increase
Change in daily revenue = ($0.15 x 40)
Change in daily revenue = $6 x 40
Change in daily revenue = $240
4. Determine the target daily revenue needed to stay in business: The company needs to take in $10,450 per day to stay in business.
5. Calculate the final fare to meet the target revenue:
Final daily revenue = Initial daily revenue + Change in daily revenue
Final daily revenue = $8000 + $240
Final daily revenue = $8240
To find the fare that will generate the target daily revenue, we divide the final daily revenue by the number of passengers:
Final daily revenue = Fare per passenger x Number of passengers
$8240 = Fare per passenger x 4000 passengers
Fare per passenger = $8240 / 4000
Fare per passenger ≈ $2.06
Therefore, the fare that should be charged is approximately $2.06 per passenger to meet the target revenue of $10,450 per day.