. In this problem, we will analyze the profit found for sales of a certain product. A demand equation (sometimes called a demand curve) shows how much money people would pay for a product depending on how much of that product is available on the open market. Often, the demand equation is found empirically (through experiment, or market research).
a. Suppose that a market research company finds that at a price of p = $31, they would sell x = 74 boxes of product each month. If they lower the price to p = $25, then more people would purchase the product, and they can expect to sell x = 80 boxes in a month’s time. Find the equation of the line for the demand equation. Write your answer in the form p = mx + b. (Hint: Write an equation using two points in the form (x,p)).
A company’s revenue is the amount of money that comes in from sales, before business costs are subtracted. For a single product, you can find the revenue by multiplying the quantity of the product sold, x, by the demand equation, p.
b. Substitute the result you found from part a into the equation R = xp to find the revenue equation. Provide your answer in simplified form.
The costs of doing business for a company can be found by adding fixed costs, such as rent, insurance, and wages, and variable costs, which are the costs to purchase the product you are selling. The portion of the company’s fixed costs allotted to this product is $320, and the supplier’s cost for materials is $3 each. Let x represent the number of product boxes.
c. If b represents a fixed cost, what value would represent b?
d. Find the cost equation for the product. Write your answer in the form C = mx + b.
The profit made from the sale of the product is found by subtracting the costs from the revenue.
e. Find the Profit Equation by substituting your equations for R and C in the equation . Simplify the equation.
f. What is the profit made from selling 24 boxes of product per month?
g. What is the profit made from selling 29 boxes of product each month?
h. What is the profit made from selling no boxes of product each month? Interpret your answer.
i. Use trial and error to find the quantity of product boxes per month that yields the highest profit.
j. How much profit would you earn from the number you found in part i?
k. At what price would you sell the product to realize this profit (hint, use the demand equation from part a)?
a. To find the equation of the line for the demand equation, we can use the equation of a line:
The two points we have are (74, 31) and (80, 25).
Using the formula (y2 - y1) / (x2 - x1) = m, we can find the slope:
(25 - 31) / (80 - 74) = -6 / 6 = -1
Using the slope-intercept form, we can substitute one of the points and the slope into the equation y = mx + b:
31 = -1 * 74 + b
31 = -74 + b
b = 31 + 74
b = 105
Therefore, the equation of the demand equation is p = -x + 105.
b. The revenue equation can be found by substituting the demand equation into the equation R = xp:
R = x * (-x + 105)
R = -x^2 + 105x
c. In this context, b represents the fixed cost. According to the problem, the fixed cost is $320. Therefore, the value of b is 320.
d. The cost equation can be found by adding the fixed cost and the variable cost per box:
C = 320 + 3x
e. To find the profit equation, we can substitute the revenue equation and the cost equation into the equation P = R - C:
P = (-x^2 + 105x) - (320 + 3x)
P = -x^2 + 105x - 320 - 3x
P = -x^2 + 102x - 320
f. To find the profit made from selling 24 boxes of product per month, substitute x = 24 into the profit equation:
P = -(24)^2 + 102(24) - 320
P = -576 + 2448 - 320
P = 1552
The profit made from selling 24 boxes of product per month is $1552.
g. To find the profit made from selling 29 boxes of product each month, substitute x = 29 into the profit equation:
P = -(29)^2 + 102(29) - 320
P = -841 + 2964 - 320
P = 1803
The profit made from selling 29 boxes of product per month is $1803.
h. To find the profit made from selling no boxes of product each month, substitute x = 0 into the profit equation:
P = -(0)^2 + 102(0) - 320
P = 0 - 0 - 320
P = -320
The profit made from selling no boxes of product each month is -$320. This means that the company would incur a loss of $320.
i. To find the quantity of product boxes per month that yields the highest profit, we need to find the vertex of the parabola represented by the profit equation. The x-coordinate of the vertex can be found using the formula x = -b / (2a), where a = -1 and b = 102:
x = -102 / (2 * (-1))
x = 102 / 2
x = 51
Therefore, the quantity of product boxes per month that yields the highest profit is 51.
j. To find the profit from the quantity of 51 product boxes per month, substitute x = 51 into the profit equation:
P = -(51)^2 + 102(51) - 320
P = -2601 + 5202 - 320
P = 2303
The profit earned from selling 51 boxes of product per month is $2303.
k. To find the price at which to sell the product to realize this profit, substitute x = 51 into the demand equation from part a:
p = -(51) + 105
p = 105 - 51
p = 54
Therefore, the price at which to sell the product to realize a profit of $2303 is $54 per box.
a. To find the equation of the demand curve, we can use the two given points: (x1, p1) = (74, 31) and (x2, p2) = (80, 25).
First, let's find the slope (m) of the line:
m = (p2 - p1) / (x2 - x1) = (25 - 31) / (80 - 74) = -6/6 = -1
Next, let's find the y-intercept (b):
Using the point-slope form of a line and substituting one of the points, we have:
p - p1 = m(x - x1)
p - 31 = -1(x - 74)
p - 31 = -x + 74
p = -x + 74 + 31
p = -x + 105
So, the equation of the demand curve is: p = -x + 105.
b. The revenue equation is given by R = xp.
Substituting the demand equation we found in part a, we have:
R = x*(-x + 105)
R = -x^2 + 105x
Therefore, the revenue equation is R = -x^2 + 105x.
c. The fixed cost is represented by the y-intercept (b) in the cost equation. Since it is given as $320, b = 320.
d. The cost equation for the product is given by C = mx + b.
We are given the supplier's cost for materials as $3 each, so m = 3.
Therefore, the cost equation is C = 3x + 320.
e. The profit equation is given by subtracting the cost equation from the revenue equation, P = R - C.
Substituting the revenue and cost equations we found in parts b and d, we have:
P = (-x^2 + 105x) - (3x + 320)
P = -x^2 + 105x - 3x - 320
P = -x^2 + 102x - 320
Therefore, the profit equation is P = -x^2 + 102x - 320.
f. To find the profit made from selling 24 boxes of product per month, substitute x = 24 into the profit equation:
P = -(24)^2 + 102(24) - 320
g. To find the profit made from selling 29 boxes of product per month, substitute x = 29 into the profit equation:
P = -(29)^2 + 102(29) - 320
h. To find the profit made from selling no boxes of product each month, substitute x = 0 into the profit equation:
P = -(0)^2 + 102(0) - 320
Interpretation: The profit is -$320, which means the company would be making a loss of $320.
i. To find the quantity of product boxes per month that yields the highest profit, we need to identify the value of x that maximizes the profit equation. This can be done by either calculating the vertex of the parabolic profit equation or by testing different x-values and finding the one that gives the highest profit.
j. Once you find the quantity of product boxes per month that yields the highest profit (let's call it x_max), substitute that value into the profit equation to find the corresponding profit value.
k. To find the price at which the product should be sold to realize the profit found in part j, substitute the x_max value into the demand equation from part a (p = -x + 105).