Question

question 7
let the demand function for a product be given by the function $d(q) = -1.55q + 220$, where $q$ is the quantity of items in demand and $d(q)$ is the price per item, in dollars, that can be charged when $q$ units are sold. suppose fixed costs of production for this item are $3,000 and variable costs are $5 per item produced. if 98 items are produced and sold, find the following:
a) the total revenue from selling 98 items (to the nearest penny).
answer: $

b) the total costs to produce 98 items (to the nearest penny).
answer: $

c) the total profits to produce 98 items (to the nearest penny. profits may or may not be negative.).
answer: $

question help:

Explanation:

Part A: Total Revenue

Step 1: Recall Revenue Formula

Revenue \( R \) is given by \( R = \text{Price} \times \text{Quantity} \). The price function is \( D(q) = -1.55q + 220 \), and quantity \( q = 98 \). First, find the price when \( q = 98 \).
\( D(98) = -1.55(98) + 220 \)

Step 2: Calculate Price

\( D(98) = -151.9 + 220 = 68.1 \) (dollars per item)

Step 3: Calculate Revenue

Revenue \( R = D(98) \times 98 = 68.1 \times 98 \)
\( R = 6673.8 \)

Step 1: Recall Cost Formula

Total Cost \( C \) is Fixed Cost \( + \) Variable Cost. Fixed Cost \( = 3000 \), Variable Cost per item \( = 5 \), Quantity \( = 98 \).
\( C = 3000 + 5 \times 98 \)

Step 2: Calculate Variable Cost

\( 5 \times 98 = 490 \)

Step 3: Calculate Total Cost

\( C = 3000 + 490 = 3490 \)

Step 1: Recall Profit Formula

Profit \( P = \text{Revenue} - \text{Cost} \). From Part A, Revenue \( = 6673.8 \), from Part B, Cost \( = 3490 \).
\( P = 6673.8 - 3490 \)

Step 2: Calculate Profit

\( P = 3183.8 \)

Answer:

\( \$6673.80 \)

Part B: Total Costs