The weekly demand for the Pulsar 40-in. high-definition television is given by the demand equation p = −0.05x + 600 (0 ≤ x ≤ 12, 000) where p denotes the wholesale unit price in dollars and x denotes the quantity demanded. The weekly total cost function associated with manufacturing these sets is given by C(x) = 0.000002x 3 − 0.03x 2 + 250x + 80, 000 where C(x) denotes the total cost incurred in producing x sets. 1. Find a revenue function R(x) that gives Pulsar’s weekly revenue in dollars when x sets are sold. Write that function out explicitly. 2. Find a profit function P(x) that gives Pulsar’s weekly revenue in dollars when x sets are sold. Write that function out explicitly. 3. Determine the level of production x (with 0 ≤ x ≤ 12, 000) that will yield the maximum profit for the manufacturer. Make sure to show that this x corresponds to an absolute maximum of profit using a strategy we have discussed. You may find it useful to use the quadratic formula. If necessary, round units to the nearest whole television set at the end of your calculations.
The weekly demand for the Pulsar 40-in. high-definition television is given by the demand equation p = −0.05x + 600 (0 ≤ x ≤ 12, 000) where p denotes the wholesale unit price in dollars and x denotes the quantity demanded. The weekly total cost function associated with manufacturing these sets is given by C(x) = 0.000002x 3 − 0.03x 2 + 250x + 80, 000 where C(x) denotes the total cost incurred in producing x sets.
1. Find a revenue function R(x) that gives Pulsar’s weekly revenue in dollars when x sets are sold. Write that function out explicitly.
2. Find a profit function P(x) that gives Pulsar’s weekly revenue in dollars when x sets are sold. Write that function out explicitly.
3. Determine the level of production x (with 0 ≤ x ≤ 12, 000) that will yield the maximum profit for the manufacturer. Make sure to show that this x corresponds to an absolute maximum of profit using a strategy we have discussed. You may find it useful to use the
4. Give the profit that corresponds to your production level from part (3). If necessary, round profit to the nearest cent.
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