Solve for d 7500=pi/6d^3

$7500 = \frac{π}{6} d^{3}$

Step 1

Rewrite the equation as $\frac{π}{6} d^{3} = 7500$ .

$\frac{π}{6} d^{3} = 7500$

Step 2

Multiply both sides of the equation by $\frac{6}{π}$ .

$\frac{6}{π} (\frac{π}{6} d^{3}) = \frac{6}{π} \cdot 7500$

Step 3

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{6}{π} (\frac{π}{6} d^{3})$ .

Combine $\frac{π}{6}$ and $d^{3}$ .

$\frac{6}{π} \cdot \frac{π d^{3}}{6} = \frac{6}{π} \cdot 7500$

Combine.

$\frac{6 (π d^{3})}{π \cdot 6} = \frac{6}{π} \cdot 7500$

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 (π d^{3})}{π \cdot 6} = \frac{6}{π} \cdot 7500$

Rewrite the expression.

$\frac{π d^{3}}{π} = \frac{6}{π} \cdot 7500$

Cancel the common factor of $π$ .

Cancel the common factor.

$\frac{π d^{3}}{π} = \frac{6}{π} \cdot 7500$

Divide $d^{3}$ by $1$ .

$d^{3} = \frac{6}{π} \cdot 7500$

Simplify the right side.

Multiply $\frac{6}{π} \cdot 7500$ .

Combine $\frac{6}{π}$ and $7500$ .

$d^{3} = \frac{6 \cdot 7500}{π}$

Multiply $6$ by $7500$ .

$d^{3} = \frac{45000}{π}$

Step 4

Take the cube root of both sides of the equation to eliminate the exponent on the left side.

$d = \sqrt[3]{\frac{45000}{π}}$

Step 5

Simplify $\sqrt[3]{\frac{45000}{π}}$ .

Rewrite $\sqrt[3]{\frac{45000}{π}}$ as $\frac{\sqrt[3]{45000}}{\sqrt[3]{π}}$ .

$d = \frac{\sqrt[3]{45000}}{\sqrt[3]{π}}$

Simplify the numerator.

Rewrite $45000$ as $10^{3} \cdot 45$ .

Factor $1000$ out of $45000$ .

$d = \frac{\sqrt[3]{1000 (45)}}{\sqrt[3]{π}}$

Rewrite $1000$ as $10^{3}$ .

$d = \frac{\sqrt[3]{10^{3} \cdot 45}}{\sqrt[3]{π}}$

Pull terms out from under the radical.

$d = \frac{10 \sqrt[3]{45}}{\sqrt[3]{π}}$

Multiply $\frac{10 \sqrt[3]{45}}{\sqrt[3]{π}}$ by $\frac{{\sqrt[3]{π}}^{2}}{{\sqrt[3]{π}}^{2}}$ .

$d = \frac{10 \sqrt[3]{45}}{\sqrt[3]{π}} \cdot \frac{{\sqrt[3]{π}}^{2}}{{\sqrt[3]{π}}^{2}}$

Combine and simplify the denominator.

Multiply $\frac{10 \sqrt[3]{45}}{\sqrt[3]{π}}$ by $\frac{{\sqrt[3]{π}}^{2}}{{\sqrt[3]{π}}^{2}}$ .

$d = \frac{10 \sqrt[3]{45} {\sqrt[3]{π}}^{2}}{\sqrt[3]{π} {\sqrt[3]{π}}^{2}}$

Raise $\sqrt[3]{π}$ to the power of $1$ .

$d = \frac{10 \sqrt[3]{45} {\sqrt[3]{π}}^{2}}{{\sqrt[3]{π}}^{1} {\sqrt[3]{π}}^{2}}$

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

$d = \frac{10 \sqrt[3]{45} {\sqrt[3]{π}}^{2}}{{\sqrt[3]{π}}^{1 + 2}}$

Add $1$ and $2$ .

$d = \frac{10 \sqrt[3]{45} {\sqrt[3]{π}}^{2}}{{\sqrt[3]{π}}^{3}}$

Rewrite ${\sqrt[3]{π}}^{3}$ as $π$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{π}$ as $π^{\frac{1}{3}}$ .

$d = \frac{10 \sqrt[3]{45} {\sqrt[3]{π}}^{2}}{{(π^{\frac{1}{3}})}^{3}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$d = \frac{10 \sqrt[3]{45} {\sqrt[3]{π}}^{2}}{π^{\frac{1}{3} \cdot 3}}$

Combine $\frac{1}{3}$ and $3$ .

$d = \frac{10 \sqrt[3]{45} {\sqrt[3]{π}}^{2}}{π^{\frac{3}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$d = \frac{10 \sqrt[3]{45} {\sqrt[3]{π}}^{2}}{π^{\frac{3}{3}}}$

Rewrite the expression.

$d = \frac{10 \sqrt[3]{45} {\sqrt[3]{π}}^{2}}{π^{1}}$

Simplify.

$d = \frac{10 \sqrt[3]{45} {\sqrt[3]{π}}^{2}}{π}$

Simplify the numerator.

Rewrite ${\sqrt[3]{π}}^{2}$ as ${(π^{2})}^{\frac{1}{3}}$ .

$d = \frac{10 \sqrt[3]{45} \sqrt[3]{π^{2}}}{π}$

Combine using the product rule for radicals.

$d = \frac{10 \sqrt[3]{π^{2} \cdot 45}}{π}$

Move $45$ to the left of $π^{2}$ .

$d = \frac{10 \sqrt[3]{45 π^{2}}}{π}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$d = \frac{10 \sqrt[3]{45 π^{2}}}{π}$

Decimal Form:

$d = 24.28590063 \dots$

Solve for d 7500=pi/6d^3

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