Calculus Examples

$(40000 + \frac{1000}{3} x) (27 - x) + (40000 + \frac{10000}{3} x) (3)$

Step 1

Simplify each term.

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Step 1.1

Combine $\frac{1000}{3}$ and $x$ .

$(40000 + \frac{1000 x}{3}) (27 - x) + (40000 + \frac{10000}{3} x) (3)$

Step 1.2

Expand $(40000 + \frac{1000 x}{3}) (27 - x)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$40000 (27 - x) + \frac{1000 x}{3} (27 - x) + (40000 + \frac{10000}{3} x) (3)$

Step 1.2.2

Apply the distributive property.

$40000 \cdot 27 + 40000 (- x) + \frac{1000 x}{3} (27 - x) + (40000 + \frac{10000}{3} x) (3)$

Step 1.2.3

Apply the distributive property.

$40000 \cdot 27 + 40000 (- x) + \frac{1000 x}{3} \cdot 27 + \frac{1000 x}{3} (- x) + (40000 + \frac{10000}{3} x) (3)$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $40000$ by $27$ .

$1080000 + 40000 (- x) + \frac{1000 x}{3} \cdot 27 + \frac{1000 x}{3} (- x) + (40000 + \frac{10000}{3} x) (3)$

Step 1.3.1.2

Multiply $- 1$ by $40000$ .

$1080000 - 40000 x + \frac{1000 x}{3} \cdot 27 + \frac{1000 x}{3} (- x) + (40000 + \frac{10000}{3} x) (3)$

Step 1.3.1.3

Cancel the common factor of $3$ .

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Step 1.3.1.3.1

Factor $3$ out of $27$ .

$1080000 - 40000 x + \frac{1000 x}{3} \cdot (3 (9)) + \frac{1000 x}{3} (- x) + (40000 + \frac{10000}{3} x) (3)$

Step 1.3.1.3.2

Cancel the common factor.

$1080000 - 40000 x + \frac{1000 x}{3} \cdot (3 \cdot 9) + \frac{1000 x}{3} (- x) + (40000 + \frac{10000}{3} x) (3)$

Step 1.3.1.3.3

Rewrite the expression.

$1080000 - 40000 x + 1000 x \cdot 9 + \frac{1000 x}{3} (- x) + (40000 + \frac{10000}{3} x) (3)$

Step 1.3.1.4

Multiply $9$ by $1000$ .

$1080000 - 40000 x + 9000 x + \frac{1000 x}{3} (- x) + (40000 + \frac{10000}{3} x) (3)$

Step 1.3.1.5

Rewrite using the commutative property of multiplication.

$1080000 - 40000 x + 9000 x - \frac{1000 x}{3} x + (40000 + \frac{10000}{3} x) (3)$

Step 1.3.1.6

Multiply $- \frac{1000 x}{3} x$ .

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Step 1.3.1.6.1

Combine $x$ and $\frac{1000 x}{3}$ .

$1080000 - 40000 x + 9000 x - \frac{x (1000 x)}{3} + (40000 + \frac{10000}{3} x) (3)$

Step 1.3.1.6.2

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

$1080000 - 40000 x + 9000 x - \frac{1000 (x^{1} x)}{3} + (40000 + \frac{10000}{3} x) (3)$

Step 1.3.1.6.3

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

$1080000 - 40000 x + 9000 x - \frac{1000 (x^{1} x^{1})}{3} + (40000 + \frac{10000}{3} x) (3)$

Step 1.3.1.6.4

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

$1080000 - 40000 x + 9000 x - \frac{1000 x^{1 + 1}}{3} + (40000 + \frac{10000}{3} x) (3)$

Step 1.3.1.6.5

Add $1$ and $1$ .

$1080000 - 40000 x + 9000 x - \frac{1000 x^{2}}{3} + (40000 + \frac{10000}{3} x) (3)$

Step 1.3.2

Add $- 40000 x$ and $9000 x$ .

$1080000 - 31000 x - \frac{1000 x^{2}}{3} + (40000 + \frac{10000}{3} x) (3)$

Step 1.4

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

$1080000 - 31000 x - \frac{1000 x^{2}}{3} + (40000 + \frac{10000 x}{3}) \cdot 3$

Step 1.5

Apply the distributive property.

$1080000 - 31000 x - \frac{1000 x^{2}}{3} + 40000 \cdot 3 + \frac{10000 x}{3} \cdot 3$

Step 1.6

Multiply $40000$ by $3$ .

$1080000 - 31000 x - \frac{1000 x^{2}}{3} + 120000 + \frac{10000 x}{3} \cdot 3$

Step 1.7

Cancel the common factor of $3$ .

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Step 1.7.1

Factor $3$ out of $3$ .

$1080000 - 31000 x - \frac{1000 x^{2}}{3} + 120000 + \frac{10000 x}{3 (1)} \cdot 3$

Step 1.7.2

Cancel the common factor.

$1080000 - 31000 x - \frac{1000 x^{2}}{3} + 120000 + \frac{10000 x}{3 \cdot 1} \cdot 3$

Step 1.7.3

Rewrite the expression.

$1080000 - 31000 x - \frac{1000 x^{2}}{3} + 120000 + 10000 x$

Step 2

Simplify by adding terms.

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Step 2.1

Add $1080000$ and $120000$ .

$- 31000 x - \frac{1000 x^{2}}{3} + 1200000 + 10000 x$

Step 2.2

Add $- 31000 x$ and $10000 x$ .

$- 21000 x - \frac{1000 x^{2}}{3} + 1200000$

Step 2.3

Reorder $- 21000 x$ and $- \frac{1000 x^{2}}{3}$ .

$- \frac{1000 x^{2}}{3} - 21000 x + 1200000$