Calculus Examples

$R (x) = (40000 + \frac{10000}{3} x) (27 - x) + (40000 + \frac{10000}{3} x) (3)$

Step 1

Combine $\frac{10000}{3}$ and $x$ .

$R (x) = (40000 + \frac{10000 x}{3}) (27 - x) + (40000 + \frac{10000}{3} x) (3)$

Step 2

Expand $(40000 + \frac{10000 x}{3}) (27 - x)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$R (x) = 40000 (27 - x) + \frac{10000 x}{3} \cdot (27 - x) + (40000 + \frac{10000}{3} x) (3)$

Step 2.2

Apply the distributive property.

$R (x) = 40000 \cdot 27 + 40000 (- x) + \frac{10000 x}{3} \cdot (27 - x) + (40000 + \frac{10000}{3} x) (3)$

Step 2.3

Apply the distributive property.

$R (x) = 40000 \cdot 27 + 40000 (- x) + \frac{10000 x}{3} \cdot 27 + \frac{10000 x}{3} \cdot (- x) + (40000 + \frac{10000}{3} x) (3)$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $40000$ by $27$ .

$R (x) = 1080000 + 40000 (- x) + \frac{10000 x}{3} \cdot 27 + \frac{10000 x}{3} \cdot (- x) + (40000 + \frac{10000}{3} x) (3)$

Step 3.1.2

Multiply $- 1$ by $40000$ .

$R (x) = 1080000 - 40000 x + \frac{10000 x}{3} \cdot 27 + \frac{10000 x}{3} \cdot (- x) + (40000 + \frac{10000}{3} x) (3)$

Step 3.1.3

Cancel the common factor of $3$ .

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Step 3.1.3.1

Factor $3$ out of $27$ .

$R (x) = 1080000 - 40000 x + \frac{10000 x}{3} \cdot (3 (9)) + \frac{10000 x}{3} \cdot (- x) + (40000 + \frac{10000}{3} x) (3)$

Step 3.1.3.2

Cancel the common factor.

$R (x) = 1080000 - 40000 x + \frac{10000 x}{3} \cdot (3 \cdot 9) + \frac{10000 x}{3} \cdot (- x) + (40000 + \frac{10000}{3} x) (3)$

Step 3.1.3.3

Rewrite the expression.

$R (x) = 1080000 - 40000 x + 10000 x \cdot 9 + \frac{10000 x}{3} \cdot (- x) + (40000 + \frac{10000}{3} x) (3)$

Step 3.1.4

Multiply $9$ by $10000$ .

$R (x) = 1080000 - 40000 x + 90000 x + \frac{10000 x}{3} \cdot (- x) + (40000 + \frac{10000}{3} x) (3)$

Step 3.1.5

Rewrite using the commutative property of multiplication.

$R (x) = 1080000 - 40000 x + 90000 x - \frac{10000 x}{3} \cdot x + (40000 + \frac{10000}{3} x) (3)$

Step 3.1.6

Multiply $- \frac{10000 x}{3} x$ .

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Step 3.1.6.1

Combine $x$ and $\frac{10000 x}{3}$ .

$R (x) = 1080000 - 40000 x + 90000 x - \frac{x (10000 x)}{3} + (40000 + \frac{10000}{3} x) (3)$

Step 3.1.6.2

Raise $x$ to the power of $1$ .

$R (x) = 1080000 - 40000 x + 90000 x - \frac{10000 (x \cdot x)}{3} + (40000 + \frac{10000}{3} x) (3)$

Step 3.1.6.3

Raise $x$ to the power of $1$ .

$R (x) = 1080000 - 40000 x + 90000 x - \frac{10000 (x \cdot x)}{3} + (40000 + \frac{10000}{3} x) (3)$

Step 3.1.6.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$R (x) = 1080000 - 40000 x + 90000 x - \frac{10000 x^{1 + 1}}{3} + (40000 + \frac{10000}{3} x) (3)$

Step 3.1.6.5

Add $1$ and $1$ .

$R (x) = 1080000 - 40000 x + 90000 x - \frac{10000 x^{2}}{3} + (40000 + \frac{10000}{3} x) (3)$

Step 3.2

Add $- 40000 x$ and $90000 x$ .

$R (x) = 1080000 + 50000 x - \frac{10000 x^{2}}{3} + (40000 + \frac{10000}{3} x) (3)$

Step 4

Combine $\frac{10000}{3}$ and $x$ .

$R (x) = 1080000 + 50000 x - \frac{10000 x^{2}}{3} + (40000 + \frac{10000 x}{3}) \cdot 3$

Step 5

Apply the distributive property.

$R (x) = 1080000 + 50000 x - \frac{10000 x^{2}}{3} + 40000 \cdot 3 + \frac{10000 x}{3} \cdot 3$

Step 6

Multiply $40000$ by $3$ .

$R (x) = 1080000 + 50000 x - \frac{10000 x^{2}}{3} + 120000 + \frac{10000 x}{3} \cdot 3$

Step 7

Cancel the common factor of $3$ .

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Step 7.1

Factor $3$ out of $3$ .

$R (x) = 1080000 + 50000 x - \frac{10000 x^{2}}{3} + 120000 + \frac{10000 x}{3 (1)} \cdot 3$

Step 7.2

Cancel the common factor.

$R (x) = 1080000 + 50000 x - \frac{10000 x^{2}}{3} + 120000 + \frac{10000 x}{3 \cdot 1} \cdot 3$

Step 7.3

Rewrite the expression.

$R (x) = 1080000 + 50000 x - \frac{10000 x^{2}}{3} + 120000 + 10000 x$

Step 8

Add $1080000$ and $120000$ .

$R (x) = 50000 x - \frac{10000 x^{2}}{3} + 1200000 + 10000 x$

Step 9

Add $50000 x$ and $10000 x$ .

$R (x) = 60000 x - \frac{10000 x^{2}}{3} + 1200000$

Step 10

Reorder terms.

$R (x) = - \frac{10000 x^{2}}{3} + 60000 x + 1200000$

Step 11

Factor $10000$ out of $- \frac{10000 x^{2}}{3} + 60000 x + 1200000$ .

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Step 11.1

Factor $10000$ out of $- \frac{10000 x^{2}}{3}$ .

$R (x) = 10000 (- \frac{x^{2}}{3}) + 60000 x + 1200000$

Step 11.2

Factor $10000$ out of $60000 x$ .

$R (x) = 10000 (- \frac{x^{2}}{3}) + 10000 (6 x) + 1200000$

Step 11.3

Factor $10000$ out of $1200000$ .

$R (x) = 10000 (- \frac{x^{2}}{3}) + 10000 (6 x) + 10000 (120)$

Step 11.4

Factor $10000$ out of $10000 (- \frac{x^{2}}{3}) + 10000 (6 x)$ .

$R (x) = 10000 (- \frac{x^{2}}{3} + 6 x) + 10000 (120)$

Step 11.5

Factor $10000$ out of $10000 (- \frac{x^{2}}{3} + 6 x) + 10000 (120)$ .

$R (x) = 10000 (- \frac{x^{2}}{3} + 6 x + 120)$