Two fraternities, Sig Ep and Ep Sig, plan to raise money jointly to benefit homeless people on Long Island. They will sell Yoda vs. Alien T-shirts in the student center, but are not sure how much to charge. Sig Ep treasurer Augustus recalls that they once sold 140 shirts in a week at $6 per shirt, but Ep Sig treasurer Julius has solid research indicating that it is possible to sell 350 per week at $3 per shirt.
(a) Based on this information, construct a linear demand equation for Yoda vs. Alien T-shirts, and hence obtain the weekly revenue R as a function of the unit price x.
R(x)=
(b) The university administration charges the fraternities a weekly fee of $500 for use of the Student Center. Find the weekly profit P as a function of the unit price x.
P(x)=
Determine how much the fraternities should charge to obtain the largest possible weekly profit.
x = $___ per T-shirt
What is the largest possible weekly profit?
How do I determine which formula to use and how to solve it?
Thank you.
Do this one just like the previous problem by "Erin".
To find the linear demand equation for the Yoda vs. Alien T-shirts, we can use the information provided by both treasurers. The demand equation can be represented as R(x), where x is the unit price of the T-shirts.
(a) To construct the demand equation, we need to find the relationship between the unit price and the number of shirts sold. Let's start with Sig Ep's sales:
- Sig Ep sold 140 shirts in one week at $6 per shirt. This gives us the point (140, 6), where 140 represents the quantity and 6 represents the price.
Next, let's consider Ep Sig's findings:
- Ep Sig's research indicates that it's possible to sell 350 shirts in one week at $3 per shirt. This gives us the point (350, 3).
To find the linear equation that connects these two points, we can use the slope-intercept form: y = mx + b, where m represents the slope and b represents the y-intercept.
First, let's find the slope (m):
m = (y2 - y1) / (x2 - x1)
m = (3 - 6) / (350 - 140)
m = -3 / 210
m = -1/70
Now, let's find the y-intercept (b) by substituting the values of either point into the slope-intercept form:
6 = (-1/70) * 140 + b
6 = -2 + b
b = 8
Therefore, the linear demand equation for the Yoda vs. Alien T-shirts is:
R(x) = (-1/70)x + 8
(b) To find the weekly profit (P) as a function of the unit price (x), we need to subtract the cost from the revenue. Since the weekly fee charged by the university is $500, we can derive the profit function by subtracting $500 from the weekly revenue R(x):
P(x) = R(x) - $500
P(x) = (-1/70)x + 8 - $500
P(x) = (-1/70)x - $492
To determine the unit price that will maximize the weekly profit, we need to find the value of x for which the profit function P(x) is at its maximum. In this case, we have a linear profit function, and the maximum value will occur at the highest price point.
To find this, we can first set the derivative of P(x) equal to zero and solve for x:
P'(x) = -1/70 = 0
Solving the above equation will give us the value of x. However, as it is a constant, it means that the function is decreasing at a constant rate. Therefore, we can conclude that there is no global maximum or minimum, and the weekly profit will continuously decrease as the unit price increases.
However, the question asks us to determine the largest possible weekly profit, which means we need to consider the upper bound of the unit price. Since the information provided does not indicate any restrictions, we can assume that the fraternities can set any price for the T-shirts. Therefore, to maximize the weekly profit, the fraternities should charge the highest possible unit price, which will be $6 per T-shirt.
Hence, the recommended charge per T-shirt for the largest possible weekly profit is $6. The largest possible weekly profit would be:
P($6) = (-1/70)(6) - $492
P($6) = -6/70 - $492
P($6) ≈ -0.086 - $492
P($6) ≈ -$492.086
Therefore, the largest possible weekly profit is approximately -$492.086.