Calculus Examples

$(40000 + \frac{10000}{3} x) (18 - x) + (40000 + \frac{10000}{3} x) (4.5)$

Step 1

Simplify each term.

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Step 1.1

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

$(40000 + \frac{10000 x}{3}) (18 - x) + (40000 + \frac{10000}{3} x) (4.5)$

Step 1.2

Expand $(40000 + \frac{10000 x}{3}) (18 - x)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$40000 (18 - x) + \frac{10000 x}{3} (18 - x) + (40000 + \frac{10000}{3} x) (4.5)$

Step 1.2.2

Apply the distributive property.

$40000 \cdot 18 + 40000 (- x) + \frac{10000 x}{3} (18 - x) + (40000 + \frac{10000}{3} x) (4.5)$

Step 1.2.3

Apply the distributive property.

$40000 \cdot 18 + 40000 (- x) + \frac{10000 x}{3} \cdot 18 + \frac{10000 x}{3} (- x) + (40000 + \frac{10000}{3} x) (4.5)$

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 $18$ .

$720000 + 40000 (- x) + \frac{10000 x}{3} \cdot 18 + \frac{10000 x}{3} (- x) + (40000 + \frac{10000}{3} x) (4.5)$

Step 1.3.1.2

Multiply $- 1$ by $40000$ .

$720000 - 40000 x + \frac{10000 x}{3} \cdot 18 + \frac{10000 x}{3} (- x) + (40000 + \frac{10000}{3} x) (4.5)$

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 $18$ .

$720000 - 40000 x + \frac{10000 x}{3} \cdot (3 (6)) + \frac{10000 x}{3} (- x) + (40000 + \frac{10000}{3} x) (4.5)$

Step 1.3.1.3.2

Cancel the common factor.

$720000 - 40000 x + \frac{10000 x}{3} \cdot (3 \cdot 6) + \frac{10000 x}{3} (- x) + (40000 + \frac{10000}{3} x) (4.5)$

Step 1.3.1.3.3

Rewrite the expression.

$720000 - 40000 x + 10000 x \cdot 6 + \frac{10000 x}{3} (- x) + (40000 + \frac{10000}{3} x) (4.5)$

Step 1.3.1.4

Multiply $6$ by $10000$ .

$720000 - 40000 x + 60000 x + \frac{10000 x}{3} (- x) + (40000 + \frac{10000}{3} x) (4.5)$

Step 1.3.1.5

Rewrite using the commutative property of multiplication.

$720000 - 40000 x + 60000 x - \frac{10000 x}{3} x + (40000 + \frac{10000}{3} x) (4.5)$

Step 1.3.1.6

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

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Step 1.3.1.6.1

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

$720000 - 40000 x + 60000 x - \frac{x (10000 x)}{3} + (40000 + \frac{10000}{3} x) (4.5)$

Step 1.3.1.6.2

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

$720000 - 40000 x + 60000 x - \frac{10000 (x^{1} x)}{3} + (40000 + \frac{10000}{3} x) (4.5)$

Step 1.3.1.6.3

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

$720000 - 40000 x + 60000 x - \frac{10000 (x^{1} x^{1})}{3} + (40000 + \frac{10000}{3} x) (4.5)$

Step 1.3.1.6.4

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

$720000 - 40000 x + 60000 x - \frac{10000 x^{1 + 1}}{3} + (40000 + \frac{10000}{3} x) (4.5)$

Step 1.3.1.6.5

Add $1$ and $1$ .

$720000 - 40000 x + 60000 x - \frac{10000 x^{2}}{3} + (40000 + \frac{10000}{3} x) (4.5)$

Step 1.3.2

Add $- 40000 x$ and $60000 x$ .

$720000 + 20000 x - \frac{10000 x^{2}}{3} + (40000 + \frac{10000}{3} x) (4.5)$

Step 1.4

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

$720000 + 20000 x - \frac{10000 x^{2}}{3} + (40000 + \frac{10000 x}{3}) \cdot 4.5$

Step 1.5

Apply the distributive property.

$720000 + 20000 x - \frac{10000 x^{2}}{3} + 40000 \cdot 4.5 + \frac{10000 x}{3} \cdot 4.5$

Step 1.6

Multiply $40000$ by $4.5$ .

$720000 + 20000 x - \frac{10000 x^{2}}{3} + 180000 + \frac{10000 x}{3} \cdot 4.5$

Step 1.7

Multiply $\frac{10000 x}{3} \cdot 4.5$ .

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Step 1.7.1

Combine $\frac{10000 x}{3}$ and $4.5$ .

$720000 + 20000 x - \frac{10000 x^{2}}{3} + 180000 + \frac{10000 x \cdot 4.5}{3}$

Step 1.7.2

Multiply $4.5$ by $10000$ .

$720000 + 20000 x - \frac{10000 x^{2}}{3} + 180000 + \frac{45000 x}{3}$

Step 1.8

Cancel the common factor of $45000$ and $3$ .

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Step 1.8.1

Factor $3$ out of $45000 x$ .

$720000 + 20000 x - \frac{10000 x^{2}}{3} + 180000 + \frac{3 (15000 x)}{3}$

Step 1.8.2

Cancel the common factors.

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Step 1.8.2.1

Factor $3$ out of $3$ .

$720000 + 20000 x - \frac{10000 x^{2}}{3} + 180000 + \frac{3 (15000 x)}{3 (1)}$

Step 1.8.2.2

Cancel the common factor.

$720000 + 20000 x - \frac{10000 x^{2}}{3} + 180000 + \frac{3 (15000 x)}{3 \cdot 1}$

Step 1.8.2.3

Rewrite the expression.

$720000 + 20000 x - \frac{10000 x^{2}}{3} + 180000 + \frac{15000 x}{1}$

Step 1.8.2.4

Divide $15000 x$ by $1$ .

$720000 + 20000 x - \frac{10000 x^{2}}{3} + 180000 + 15000 x$

Step 2

Simplify by adding terms.

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Step 2.1

Add $720000$ and $180000$ .

$20000 x - \frac{10000 x^{2}}{3} + 900000 + 15000 x$

Step 2.2

Add $20000 x$ and $15000 x$ .

$35000 x - \frac{10000 x^{2}}{3} + 900000$

Step 2.3

Reorder $35000 x$ and $- \frac{10000 x^{2}}{3}$ .

$- \frac{10000 x^{2}}{3} + 35000 x + 900000$