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Business applications: cost, revenue and profit practice problems

Cost, revenue and profit: marginal cost, and the production level that maximizes profit. 60 problems with worked solutions; in 60 of them every equation is proved by a computer algebra system.

1 worked problems

Try each one before opening the solution. Every line is labeled with what checked it. How we verify

Problem 3.650

medium

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{200} + 4 x + 1500 \) dollars, and each unit sells for \( \displaystyle \$64 \). Find the marginal cost at \( \displaystyle x = 50 \), and the number of units that maximizes profit, and that profit.

\[ \left. \frac{d}{d x} \left(\frac{x^{2}}{200} + 4 x + 1500\right) \right|_{\substack{ x=50 }} = \frac{9}{2} \]
Marginal cost C′(50): about the cost of unit 51.✓ Proved
\[ 64 x - \left(\frac{x^{2}}{200} + 4 x + 1500\right) = - \frac{x^{2}}{200} + 60 x - 1500 \]
Profit = revenue − cost.✓ Proved
\[ \frac{d}{d x} \left(- \frac{x^{2}}{200} + 60 x - 1500\right) = 60 - \frac{x}{100} \]
P′(x).✓ Proved
\[ \left. 60 - \frac{x}{100} \right|_{\substack{ x=6000 }} = 0 \]
P′ = 0 at x = 6000.✓ Proved
\[ \frac{d^{2}}{d x^{2}} \left(- \frac{x^{2}}{200} + 60 x - 1500\right) = - \frac{1}{100} \]
P″ < 0: a maximum.✓ Proved
\[ \left. 60 x + \frac{\left(-1\right) x^{2}}{200} - 1500 \right|_{\substack{ x=6000 }} = 178500 \]
The maximum profit.✓ Proved

Answer \( C'(50) = \frac{9}{2},\ x = 6000,\ P(6000) = 178500 \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

More business applications: cost, revenue and profit practice

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{50} + 8 x + 200 \) dollars, and each unit sells for \( \displaystyle \$28 \). Find the marginal cost at \( \displaystyle x = 10 \), and the number of units that maximizes profit, and that profit.

Problem 3.651medium✓ Every equation proved

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{20} + 2 x + 1000 \) dollars, and each unit sells for \( \displaystyle \$32 \). Find the marginal cost at \( \displaystyle x = 100 \), and the number of units that maximizes profit, and that profit.

Problem 3.652medium✓ Nihil obstat

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{20} + 3 x + 200 \) dollars, and each unit sells for \( \displaystyle \$63 \). Find the marginal cost at \( \displaystyle x = 100 \), and the number of units that maximizes profit, and that profit.

Problem 3.653medium✓ Nihil obstat

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{10} + 8 x + 500 \) dollars, and each unit sells for \( \displaystyle \$28 \). Find the marginal cost at \( \displaystyle x = 20 \), and the number of units that maximizes profit, and that profit.

Problem 3.654medium✓ Nihil obstat

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{200} + 5 x + 500 \) dollars, and each unit sells for \( \displaystyle \$25 \). Find the marginal cost at \( \displaystyle x = 10 \), and the number of units that maximizes profit, and that profit.

Problem 3.655medium✓ Every equation proved

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{100} + 4 x + 1000 \) dollars, and each unit sells for \( \displaystyle \$14 \). Find the marginal cost at \( \displaystyle x = 20 \), and the number of units that maximizes profit, and that profit.

Problem 3.656medium✓ Nihil obstat

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{50} + 8 x + 100 \) dollars, and each unit sells for \( \displaystyle \$68 \). Find the marginal cost at \( \displaystyle x = 10 \), and the number of units that maximizes profit, and that profit.

Problem 3.657medium✓ Every equation proved

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{200} + 10 x + 100 \) dollars, and each unit sells for \( \displaystyle \$70 \). Find the marginal cost at \( \displaystyle x = 100 \), and the number of units that maximizes profit, and that profit.

Problem 3.658medium✓ Nihil obstat

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{200} + 3 x + 500 \) dollars, and each unit sells for \( \displaystyle \$63 \). Find the marginal cost at \( \displaystyle x = 10 \), and the number of units that maximizes profit, and that profit.

Problem 3.659medium✓ Every equation proved

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{100} + 8 x + 1000 \) dollars, and each unit sells for \( \displaystyle \$68 \). Find the marginal cost at \( \displaystyle x = 20 \), and the number of units that maximizes profit, and that profit.

Problem 3.660medium✓ Nihil obstat

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{20} + 10 x + 100 \) dollars, and each unit sells for \( \displaystyle \$30 \). Find the marginal cost at \( \displaystyle x = 100 \), and the number of units that maximizes profit, and that profit.

Problem 3.661medium✓ Nihil obstat

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{50} + 5 x + 100 \) dollars, and each unit sells for \( \displaystyle \$45 \). Find the marginal cost at \( \displaystyle x = 10 \), and the number of units that maximizes profit, and that profit.

Problem 3.662medium✓ Every equation proved

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{200} + 5 x + 200 \) dollars, and each unit sells for \( \displaystyle \$25 \). Find the marginal cost at \( \displaystyle x = 10 \), and the number of units that maximizes profit, and that profit.

Problem 3.663medium✓ Nihil obstat

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{50} + 10 x + 100 \) dollars, and each unit sells for \( \displaystyle \$20 \). Find the marginal cost at \( \displaystyle x = 50 \), and the number of units that maximizes profit, and that profit.

Problem 3.664medium✓ Nihil obstat

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{200} + 10 x + 200 \) dollars, and each unit sells for \( \displaystyle \$20 \). Find the marginal cost at \( \displaystyle x = 50 \), and the number of units that maximizes profit, and that profit.

Problem 3.680medium✓ Nihil obstat

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{200} + 5 x + 100 \) dollars, and each unit sells for \( \displaystyle \$65 \). Find the marginal cost at \( \displaystyle x = 10 \), and the number of units that maximizes profit, and that profit.

Problem 3.681medium✓ Every equation proved

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{20} + 2 x + 1000 \) dollars, and each unit sells for \( \displaystyle \$22 \). Find the marginal cost at \( \displaystyle x = 50 \), and the number of units that maximizes profit, and that profit.

Problem 3.682medium✓ Nihil obstat

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{200} + 2 x + 1000 \) dollars, and each unit sells for \( \displaystyle \$42 \). Find the marginal cost at \( \displaystyle x = 10 \), and the number of units that maximizes profit, and that profit.

Problem 3.683medium✓ Every equation proved

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{10} + 8 x + 500 \) dollars, and each unit sells for \( \displaystyle \$18 \). Find the marginal cost at \( \displaystyle x = 50 \), and the number of units that maximizes profit, and that profit.

Problem 3.684medium✓ Nihil obstat

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{50} + 10 x + 500 \) dollars, and each unit sells for \( \displaystyle \$50 \). Find the marginal cost at \( \displaystyle x = 100 \), and the number of units that maximizes profit, and that profit.

Problem 3.685medium✓ Nihil obstat

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{50} + 8 x + 500 \) dollars, and each unit sells for \( \displaystyle \$18 \). Find the marginal cost at \( \displaystyle x = 10 \), and the number of units that maximizes profit, and that profit.

Problem 3.686medium✓ Every equation proved

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{50} + 8 x + 500 \) dollars, and each unit sells for \( \displaystyle \$18 \). Find the marginal cost at \( \displaystyle x = 50 \), and the number of units that maximizes profit, and that profit.

Problem 3.687medium✓ Nihil obstat

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{200} + 2 x + 1000 \) dollars, and each unit sells for \( \displaystyle \$22 \). Find the marginal cost at \( \displaystyle x = 100 \), and the number of units that maximizes profit, and that profit.

Problem 3.688medium✓ Nihil obstat

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{200} + 5 x + 1000 \) dollars, and each unit sells for \( \displaystyle \$65 \). Find the marginal cost at \( \displaystyle x = 10 \), and the number of units that maximizes profit, and that profit.

Problem 3.689medium✓ Every equation proved

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{100} + 2 x + 100 \) dollars, and each unit sells for \( \displaystyle \$42 \). Find the marginal cost at \( \displaystyle x = 20 \), and the number of units that maximizes profit, and that profit.

Problem 3.705medium✓ Nihil obstat

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{50} + 4 x + 100 \) dollars, and each unit sells for \( \displaystyle \$14 \). Find the marginal cost at \( \displaystyle x = 10 \), and the number of units that maximizes profit, and that profit.

Problem 3.706medium✓ Every equation proved

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{20} + 3 x + 200 \) dollars, and each unit sells for \( \displaystyle \$23 \). Find the marginal cost at \( \displaystyle x = 50 \), and the number of units that maximizes profit, and that profit.

Problem 3.707medium✓ Nihil obstat

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{200} + 4 x + 100 \) dollars, and each unit sells for \( \displaystyle \$34 \). Find the marginal cost at \( \displaystyle x = 50 \), and the number of units that maximizes profit, and that profit.

Problem 3.708medium✓ Nihil obstat

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{200} + 8 x + 1500 \) dollars, and each unit sells for \( \displaystyle \$48 \). Find the marginal cost at \( \displaystyle x = 100 \), and the number of units that maximizes profit, and that profit.

Problem 3.709medium✓ Nihil obstat

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{200} + 3 x + 200 \) dollars, and each unit sells for \( \displaystyle \$33 \). Find the marginal cost at \( \displaystyle x = 100 \), and the number of units that maximizes profit, and that profit.

Problem 3.710medium✓ Nihil obstat

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{20} + 4 x + 200 \) dollars, and each unit sells for \( \displaystyle \$64 \). Find the marginal cost at \( \displaystyle x = 20 \), and the number of units that maximizes profit, and that profit.

Problem 3.711medium✓ Every equation proved

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{100} + 2 x + 500 \) dollars, and each unit sells for \( \displaystyle \$32 \). Find the marginal cost at \( \displaystyle x = 20 \), and the number of units that maximizes profit, and that profit.

Problem 3.712medium✓ Nihil obstat

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{100} + 2 x + 100 \) dollars, and each unit sells for \( \displaystyle \$62 \). Find the marginal cost at \( \displaystyle x = 50 \), and the number of units that maximizes profit, and that profit.

Problem 3.713medium✓ Nihil obstat

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{200} + 5 x + 500 \) dollars, and each unit sells for \( \displaystyle \$25 \). Find the marginal cost at \( \displaystyle x = 50 \), and the number of units that maximizes profit, and that profit.

Problem 3.714medium✓ Nihil obstat

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{200} + 3 x + 1000 \) dollars, and each unit sells for \( \displaystyle \$63 \). Find the marginal cost at \( \displaystyle x = 20 \), and the number of units that maximizes profit, and that profit.

Problem 3.715medium✓ Nihil obstat

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{20} + 5 x + 100 \) dollars, and each unit sells for \( \displaystyle \$65 \). Find the marginal cost at \( \displaystyle x = 100 \), and the number of units that maximizes profit, and that profit.

Problem 3.716medium✓ Nihil obstat

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{50} + 4 x + 200 \) dollars, and each unit sells for \( \displaystyle \$14 \). Find the marginal cost at \( \displaystyle x = 10 \), and the number of units that maximizes profit, and that profit.

Problem 3.717medium✓ Nihil obstat

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{20} + 3 x + 500 \) dollars, and each unit sells for \( \displaystyle \$43 \). Find the marginal cost at \( \displaystyle x = 10 \), and the number of units that maximizes profit, and that profit.

Problem 3.718medium✓ Every equation proved

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{50} + 5 x + 1000 \) dollars, and each unit sells for \( \displaystyle \$45 \). Find the marginal cost at \( \displaystyle x = 50 \), and the number of units that maximizes profit, and that profit.

Problem 3.719medium✓ Nihil obstat

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{200} + 10 x + 1000 \) dollars, and each unit sells for \( \displaystyle \$70 \). Find the marginal cost at \( \displaystyle x = 100 \), and the number of units that maximizes profit, and that profit.

Problem 3.729medium✓ Every equation proved

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{50} + 8 x + 200 \) dollars, and each unit sells for \( \displaystyle \$38 \). Find the marginal cost at \( \displaystyle x = 20 \), and the number of units that maximizes profit, and that profit.

Problem 3.730medium✓ Nihil obstat

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{10} + 8 x + 200 \) dollars, and each unit sells for \( \displaystyle \$28 \). Find the marginal cost at \( \displaystyle x = 100 \), and the number of units that maximizes profit, and that profit.

Problem 3.731medium✓ Nihil obstat

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{200} + 10 x + 1000 \) dollars, and each unit sells for \( \displaystyle \$40 \). Find the marginal cost at \( \displaystyle x = 20 \), and the number of units that maximizes profit, and that profit.

Problem 3.732medium✓ Every equation proved

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{20} + 3 x + 200 \) dollars, and each unit sells for \( \displaystyle \$43 \). Find the marginal cost at \( \displaystyle x = 10 \), and the number of units that maximizes profit, and that profit.

Problem 3.733medium✓ Nihil obstat

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{50} + 8 x + 1000 \) dollars, and each unit sells for \( \displaystyle \$28 \). Find the marginal cost at \( \displaystyle x = 100 \), and the number of units that maximizes profit, and that profit.

Problem 3.734medium✓ Nihil obstat

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{20} + 8 x + 1500 \) dollars, and each unit sells for \( \displaystyle \$38 \). Find the marginal cost at \( \displaystyle x = 100 \), and the number of units that maximizes profit, and that profit.

Problem 3.735medium✓ Nihil obstat

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{10} + 4 x + 500 \) dollars, and each unit sells for \( \displaystyle \$34 \). Find the marginal cost at \( \displaystyle x = 100 \), and the number of units that maximizes profit, and that profit.

Problem 3.736medium✓ Every equation proved

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{50} + 3 x + 500 \) dollars, and each unit sells for \( \displaystyle \$33 \). Find the marginal cost at \( \displaystyle x = 100 \), and the number of units that maximizes profit, and that profit.

Problem 3.737medium✓ Nihil obstat

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{20} + 8 x + 100 \) dollars, and each unit sells for \( \displaystyle \$28 \). Find the marginal cost at \( \displaystyle x = 100 \), and the number of units that maximizes profit, and that profit.

Problem 3.738medium✓ Nihil obstat

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{100} + 5 x + 200 \) dollars, and each unit sells for \( \displaystyle \$45 \). Find the marginal cost at \( \displaystyle x = 50 \), and the number of units that maximizes profit, and that profit.

Problem 3.750medium✓ Nihil obstat

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{100} + 10 x + 200 \) dollars, and each unit sells for \( \displaystyle \$50 \). Find the marginal cost at \( \displaystyle x = 100 \), and the number of units that maximizes profit, and that profit.

Problem 3.751medium✓ Nihil obstat

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{100} + 10 x + 500 \) dollars, and each unit sells for \( \displaystyle \$30 \). Find the marginal cost at \( \displaystyle x = 50 \), and the number of units that maximizes profit, and that profit.

Problem 3.752medium✓ Nihil obstat

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{200} + 10 x + 500 \) dollars, and each unit sells for \( \displaystyle \$20 \). Find the marginal cost at \( \displaystyle x = 100 \), and the number of units that maximizes profit, and that profit.

Problem 3.753medium✓ Nihil obstat

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{100} + 3 x + 200 \) dollars, and each unit sells for \( \displaystyle \$43 \). Find the marginal cost at \( \displaystyle x = 50 \), and the number of units that maximizes profit, and that profit.

Problem 3.754medium✓ Every equation proved

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{20} + 4 x + 500 \) dollars, and each unit sells for \( \displaystyle \$44 \). Find the marginal cost at \( \displaystyle x = 20 \), and the number of units that maximizes profit, and that profit.

Problem 3.755medium✓ Nihil obstat

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{50} + 8 x + 1000 \) dollars, and each unit sells for \( \displaystyle \$38 \). Find the marginal cost at \( \displaystyle x = 100 \), and the number of units that maximizes profit, and that profit.

Problem 3.756medium✓ Nihil obstat

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{200} + 2 x + 100 \) dollars, and each unit sells for \( \displaystyle \$62 \). Find the marginal cost at \( \displaystyle x = 100 \), and the number of units that maximizes profit, and that profit.

Problem 3.757medium✓ Nihil obstat

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{100} + 10 x + 1000 \) dollars, and each unit sells for \( \displaystyle \$70 \). Find the marginal cost at \( \displaystyle x = 10 \), and the number of units that maximizes profit, and that profit.

Problem 3.758medium✓ Nihil obstat

A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{200} + 2 x + 1500 \) dollars, and each unit sells for \( \displaystyle \$42 \). Find the marginal cost at \( \displaystyle x = 100 \), and the number of units that maximizes profit, and that profit.

Problem 3.759medium✓ Nihil obstat