BoxedNGone truck rentals calculates that its price function is p(x)=160-2x,where p is the price (in dollars) at which exactly x trucks will be rented per day. Find the number of trucks that BoxedNGone should rent and the price it should charge to maximize revenue. Also find the maximum revenue.
quantity=?
price=?
revenue=?
Well, if we want to find the number of trucks that should be rented to maximize revenue, we need to find the maximum point on the revenue function. Revenue is calculated by multiplying the quantity (number of trucks rented) by the price.
So, the revenue function can be expressed as R(x) = x * p(x), where p(x) is the price function given as p(x) = 160 - 2x.
Let's substitute the price function into the revenue function:
R(x) = x * (160 - 2x)
R(x) = 160x - 2x^2
To find the maximum revenue, we need to find the value of x that will make R(x) as large as possible. One way to do this is by finding the vertex of the quadratic function.
The x-coordinate of the vertex can be found using the formula: x = -b / (2a), where a = -2 and b = 160.
x = -160 / (2 * -2)
x = -160 / -4
x = 40
Now we know that the quantity (number of trucks rented) should be 40.
To find the price, we can substitute the value of x back into the price function:
p(x) = 160 - 2x
p(40) = 160 - 2 * 40
p(40) = 160 - 80
p(40) = 80
So, BoxedNGone should rent 40 trucks and charge $80 per truck to maximize revenue.
To calculate the maximum revenue, we can substitute the value of x back into the revenue function:
R(x) = x * p(x)
R(40) = 40 * (160 - 2 * 40)
R(40) = 40 * (160 - 80)
R(40) = 40 * 80
R(40) = 3200
Therefore, the maximum revenue for BoxedNGone truck rentals is $3200.
quantity = 40 trucks
price = $80 per truck
revenue = $3200
To determine the number of trucks that BoxedNGone should rent and the price it should charge to maximize revenue, and to find the maximum revenue, we need to find the vertex of the quadratic function that represents the revenue.
The revenue function can be calculated by multiplying the quantity of trucks rented (x) by the price per truck (p):
Revenue = x * p(x)
Given p(x) = 160 - 2x, we can substitute this into the revenue function:
Revenue = x * (160 - 2x)
To find the vertex, we need to convert the revenue function into vertex form. We can do this by completing the square:
Revenue = x * (160 - 2x)
= -2x^2 + 160x
Factoring out a -2 from the expression:
Revenue = -2(x^2 - 80x)
Completing the square by adding and subtracting (80/2)^2 = 1600:
Revenue = -2(x^2 - 80x + 1600 - 1600)
= -2((x - 40)^2 - 1600)
Expanding:
Revenue = -2(x - 40)^2 + 3200
Now we have the revenue function in vertex form, Revenue = a(x - h)^2 + k, where the vertex is located at (h, k).
Comparing this to the given revenue function, we can see that the vertex is located at (40, 3200).
Therefore, the quantity of trucks that BoxedNGone should rent to maximize revenue is 40.
To find the price that BoxedNGone should charge to maximize the revenue, we substitute the quantity of trucks into the price function:
p(x) = 160 - 2x
= 160 - 2(40)
= 160 - 80
= 80
Therefore, the price that BoxedNGone should charge to maximize revenue is $80 per truck.
To find the maximum revenue, we substitute the quantity of trucks into the revenue function:
Revenue = -2(x - 40)^2 + 3200
= -2(40 - 40)^2 + 3200
= -2(0) + 3200
= 3200
Therefore, the maximum revenue that BoxedNGone can achieve is $3200.
To find the number of trucks that BoxedNGone should rent and the price it should charge to maximize revenue, we need to first understand the concept of revenue and how it relates to the price and quantity.
Revenue is the total amount of money earned from selling a certain quantity of goods at a given price. It is calculated by multiplying the price per unit by the quantity sold.
In this case, the price function is given as p(x) = 160 - 2x, where p is the price in dollars and x is the number of trucks rented per day.
To find the number of trucks that BoxedNGone should rent and the price it should charge to maximize revenue, we need to find the maximum point of the revenue function.
The revenue function, R(x), can be calculated by multiplying the price function by the quantity (x):
R(x) = x * p(x)
= x * (160 - 2x)
To find the maximum revenue, we need to find the value of x that maximizes the revenue function R(x).
To do this, we can take the derivative of the revenue function with respect to x and set it equal to zero:
R'(x) = 0
Let's differentiate R(x) with respect to x:
R'(x) = (160 - 2x) + x * (-2)
Simplifying this expression:
R'(x) = 160 - 2x - 2x
= 160 - 4x
Setting R'(x) equal to zero:
160 - 4x = 0
Now, solve for x:
4x = 160
x = 160/4
x = 40
So, the number of trucks that BoxedNGone should rent to maximize revenue is 40.
To find the price that BoxedNGone should charge, we can substitute the value of x back into the price function:
p(x) = 160 - 2x
p(40) = 160 - 2 * 40
p(40) = 160 - 80
p(40) = 80
So, the price that BoxedNGone should charge to maximize revenue is $80.
To find the maximum revenue, substitute the value of x back into the revenue function:
R(x) = x * p(x)
R(40) = 40 * p(40)
R(40) = 40 * 80
R(40) = 3200
Therefore, the maximum revenue that BoxedNGone can earn is $3200.
revenue = price * quantity, so
r = x(160-2x) = 160x - 2x^2
max revenue when r'=0, or x = 40
So, just plug that in as needed.