For the cost and price functions below, find the number, q, of units that produces maximum profit; the price, p, per unit that produces maximum profit; and the maximum profit, P.
C(q)=90+16q; p=64-2q
The profit function is given by:
P(q) = pq - C(q) = (64-2q)q - (90+16q) = -2q^2 + 32q - 90
To find the number of units that produces maximum profit, we need to find the vertex of the parabola. The vertex occurs at:
q = -b/2a = -32/(-4) = 8
Therefore, the optimum number of units is 8.
To find the price per unit that produces maximum profit, we substitute q=8 into the price function:
p = 64 - 2q = 64 - 16 = 48
Therefore, the optimum price per unit is 48.
Finally, to find the maximum profit, we substitute q=8 and p=48 into the profit function:
P = pq - C(q) = (48)(8) - (90+16(8)) = $114
Therefore, the maximum profit is $114.
An accountant for a corporation forgot to pay the firm's income tax of $725,933.62 on time. The government charged a penalty of 8.3% interest for the 55 days the money was late. Find the total amount (tax and penalty) that was paid. Assume 365 days in a year.
First, we need to find the amount of penalty charged:
Penalty = Tax amount x Rate x Time
Penalty = 725,933.62 x 0.083 x (55/365)
Penalty = 9,627.28
The penalty charged was $9,627.28.
To find the total amount (tax and penalty) paid, we add the penalty to the original tax amount:
Total amount = Tax amount + Penalty
Total amount = 725,933.62 + 9,627.28
Total amount = $735,560.90
Therefore, the total amount (tax and penalty) that was paid was $735,560.90.
To find the number of units that produces maximum profit, as well as the corresponding price per unit and the maximum profit itself, we need to understand that profit is calculated as revenue minus cost.
The revenue is obtained by multiplying the number of units sold (q) by the price per unit (p), so it can be expressed as R(q) = q * p.
The cost function is given as C(q) = 90 + 16q.
To find the profit function, P(q), we subtract the cost function from the revenue function: P(q) = R(q) - C(q).
Substituting the given price function into the revenue function, we have R(q) = q * (64 - 2q).
Now, we can express the profit function: P(q) = q * (64 - 2q) - (90 + 16q).
To maximize profit, we need to find the rate of change of the profit function with respect to q (dP/dq) and set it equal to zero. Then we solve for q.
Let's calculate dP/dq:
dP/dq = 64 - 2q - 16
Setting dP/dq equal to zero:
0 = 64 - 2q - 16
Rearrange to solve for q:
2q = 64 - 16
2q = 48
q = 24
So, the number of units that produces maximum profit is 24.
To find the corresponding price per unit, we substitute q = 24 into the price function p = 64 - 2q:
p = 64 - 2 * 24
p = 64 - 48
p = 16
Therefore, the price per unit that produces maximum profit is 16.
To find the maximum profit, we substitute q = 24 and p = 16 into the profit function P(q):
P(24) = 24 * (64 - 2 * 24) - (90 + 16 * 24)
P(24) = 24 * (64 - 48) - (90 + 384)
P(24) = 24 * 16 - 474
P(24) = 384 - 474
P(24) = -90
The maximum profit is -90.