A manufacturer has determined that pro�t P (in dollars) is a quadratic function of dollars per month
a spent on advertising. The relationship is given by
p=1+50a-0.1a^2
if a is at most 500 dollars per month
-How much should be spent on advertising if pro�t is to be a maximum?
-How much should be spent on advertising if a pro�t of $5000 is desired?
-The manufacturer is spending $350 per month on advertising. Should the manufacturer increase
or decrease that amount? Explain.
a) where is the vertex?
.1 a^2 - 50 a -1 = -p
a^2 -500 a -10 = -10 p
a^2 - 500 a = -10 p + 10
a^2 - 500 a + 250^2 = -10 p + 62510
(a-250)^2 = -10 (p-6251
so
max profit of 6251 at a = 250
5000 = 1+ 50 a -.1 a^2
.1 a^2 - 50 a - 1 = -5000
.1 a^2 - 50 a + 4999 = 0
solve quadratic roots are 250+/-3sqrt1390
250 +/- 112
so
362 or 138
max profit is at 250 so if spending more than that, spend less
To find the amount that should be spent on advertising to maximize profit, we can use the fact that profit is a quadratic function. The formula for profit is given by:
P = 1 + 50a - 0.1a^2
To find the amount that maximizes profit, we can take the derivative of the profit function with respect to a, set it equal to zero, and solve for a.
1. To find the amount that should be spent on advertising if profit is to be a maximum, let's find the derivative of the profit function:
dP/da = 50 - 0.2a
Next, set the derivative equal to zero and solve for a:
50 - 0.2a = 0
0.2a = 50
a = 250
Therefore, the manufacturer should spend $250 on advertising if profit is to be maximized.
2. To find the amount that should be spent on advertising if a profit of $5000 is desired, we can set the profit function equal to $5000 and solve for a:
5000 = 1 + 50a - 0.1a^2
This equation is quadratic, so we can rearrange it and solve for a using the quadratic formula:
0.1a^2 - 50a + 4999 = 0
Using the quadratic formula, we get:
a ≈ 49.287 or a ≈ 100.713
So, the manufacturer should spend approximately $49.29 or $100.71 on advertising to achieve a profit of $5000.
3. To determine whether the manufacturer should increase or decrease the advertising budget of $350 per month, we can compare the profit at this spending level with the profit at a nearby value.
For example, we can calculate the profit at $300 per month:
P = 1 + 50(300) - 0.1(300)^2
P ≈ 1 + 15000 - 9000
P ≈ 6101
And we can calculate the profit at $400 per month:
P = 1 + 50(400) - 0.1(400)^2
P ≈ 1 + 20000 - 16000
P ≈ 4101
Comparing these profits, we can see that the profit decreases from $4101 to $6101 as the advertising budget decreases from $400 to $300.
Therefore, the manufacturer should increase the advertising budget from $350 to maximize profit.