The monthly revenue achieved by selling x wristwatches is figured to be x(40-0.2x)dollars. The wholesale cost of each watch is $32.
a. How many watches need to be sold each month to achieve a profit (revenue-cost) of $50?
for this part I did
x(40-.2x)-32x=50
I got 99.5 so I rounded it to 100
I'm not sure if a is right and I can't figure how to do b and c.
Perhaps for b you do the vertex -b/2a?
but I get 30 which can't be right.
b. What is the maximum revenue this firm can earn?
c. How many wristwatches should the firm sell to maximize profit?
first of all, 100 can't be right.
100(40-20) - 32(100) = -1200 and not 50
You would be solving
-.2x^2 +8x - 50 = 0
multiply by -5
x^2 - 40x + 250 = 0
x = 32.25 or x = 7.75 OR 32 or 8
check: if x=8
8(40-.2(8)) - 32(8) = 51.2 close enough
if x = 32
32(40-.2(32)) - 32(32) = 51.2 , ok
Your profit equation is
Profit = x(40 - .2x) - 32x
= -.2x^2 + 8x
You are probably studying the parabola.
isn't the above a parabola opening downwards?
Wouldn't the vertex give you all the information you need?
What method have you learned to find the vertex ?
Perhaps for b you do the vertex -b/2a?
but I get 30 is that right?
I can't understand how part a. could be either 8 or 32. How could it be more than one answer?
if your profit function is
-.2x^2 + 8x
then the value of -b/(2a) = -8/(2(-.2)) = 20 (you had 30)
then if x=20
maximum profit = 20(40 - .2(20) - 32(20) = 80
As to your last question, doesn't every parabola have the same y value for 2 different x values, except the x value of the vertex ?
Didn't I show above that both 8 and 32 produce a profit of appr. 50 ??
To solve this problem, let's go step by step.
a. To find how many watches need to be sold each month to achieve a profit of $50, you correctly set up the equation:
x(40 - 0.2x) - 32x = 50
To solve this equation, we can use algebraic methods. Simplify the equation by distributing x:
40x - 0.2x^2 - 32x = 50
Combine like terms:
7.8x - 0.2x^2 = 50
Rearrange the equation into a quadratic form:
0.2x^2 - 7.8x + 50 = 0
Now, to solve this quadratic equation, you can use the quadratic formula or complete the square method. I'll use the quadratic formula:
x = [-b ± √(b^2 - 4ac)] / (2a)
Here, a = 0.2, b = -7.8, and c = 50. Plug these values into the formula:
x = [-(-7.8) ± √((-7.8)^2 - 4(0.2)(50))] / (2 * 0.2)
Simplifying further:
x = [7.8 ± √(60.84 - 40)] / 0.4
x = [7.8 ± √20.84] / 0.4
Now, calculate the two possible solutions:
x1 = (7.8 + √20.84) / 0.4
x2 = (7.8 - √20.84) / 0.4
x1 ≈ 30.48
x2 ≈ 4.02
Since selling 4.02 watches doesn't make sense in this context, we can conclude that approximately 30.48 watches need to be sold each month to achieve a profit of $50. However, since we're dealing with a whole number, we round it up to 31 watches.
b. To find the maximum revenue this firm can earn, you correctly mentioned using the vertex formula:
The vertex formula states that the x-value of the vertex is given by -b/2a.
In this case, a = 0.2 and b = -7.8. Plug these values into the formula:
x = -(-7.8) / 2 * 0.2
x = 7.8 / 0.4
x ≈ 19.5
To find the maximum revenue, substitute x ≈ 19.5 into the revenue equation:
Revenue = x(40 - 0.2x)
Revenue ≈ 19.5(40 - 0.2 * 19.5)
Revenue ≈ 19.5(40 - 3.9)
Revenue ≈ 19.5(36.1)
Revenue ≈ 703.95
Therefore, the maximum revenue this firm can earn is approximately $703.95.
c. To determine how many wristwatches the firm should sell to maximize profit, we need to find the x-value at which the profit is maximized. This can be done by locating the vertex of the profit function.
The profit function can be obtained by subtracting the cost from the revenue:
Profit = Revenue - Cost
Profit = x(40 - 0.2x) - 32x
To find the x-value of the vertex, you're correct in using the formula -b/2a. In this case, a = -0.2 and b = 40 - 32:
x = -(40 - 32)/ 2 * -0.2
x = -8 / -0.4
x = 20
So, the firm should sell approximately 20 wristwatches to maximize profit.