Solve for F F*(cos(30))=500

$F \cdot (\cos (30)) = 500$

Step 1

Divide each term in $(F \cdot (\cos (30))) = 500$ by $\cos (30)$ .

$\frac{(F \cdot (\cos (30)))}{\cos (30)} = \frac{500}{\cos (30)}$

Step 2

Simplify the left side.

Separate fractions.

$\frac{(F \cdot (\cos (30)))}{1} \cdot \frac{1}{\cos (30)} = \frac{500}{\cos (30)}$

Convert from $\frac{1}{\cos (30)}$ to $\sec (30)$ .

$\frac{(F \cdot (\cos (30)))}{1} \sec (30) = \frac{500}{\cos (30)}$

Divide $(F \cdot (\cos (30)))$ by $1$ .

$(F \cdot (\cos (30))) \sec (30) = \frac{500}{\cos (30)}$

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$(\frac{F \sqrt{3}}{2}) \sec (30) = \frac{500}{\cos (30)}$

The exact value of $\sec (30)$ is $\frac{2}{\sqrt{3}}$ .

$(\frac{F \sqrt{3}}{2}) \frac{2}{\sqrt{3}} = \frac{500}{\cos (30)}$

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$(\frac{F \sqrt{3}}{2}) (\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}) = \frac{500}{\cos (30)}$

Combine fractions.

Combine and simplify the denominator.

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$(\frac{F \sqrt{3}}{2}) \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}} = \frac{500}{\cos (30)}$

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

$(\frac{F \sqrt{3}}{2}) \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} = \frac{500}{\cos (30)}$

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

$(\frac{F \sqrt{3}}{2}) \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} = \frac{500}{\cos (30)}$

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

$(\frac{F \sqrt{3}}{2}) \frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}} = \frac{500}{\cos (30)}$

Add $1$ and $1$ .

$(\frac{F \sqrt{3}}{2}) \frac{2 \sqrt{3}}{{\sqrt{3}}^{2}} = \frac{500}{\cos (30)}$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$(\frac{F \sqrt{3}}{2}) \frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} = \frac{500}{\cos (30)}$

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

$(\frac{F \sqrt{3}}{2}) \frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}} = \frac{500}{\cos (30)}$

Combine $\frac{1}{2}$ and $2$ .

$(\frac{F \sqrt{3}}{2}) \frac{2 \sqrt{3}}{3^{\frac{2}{2}}} = \frac{500}{\cos (30)}$

Cancel the common factor of $2$ .

Cancel the common factor.

$(\frac{F \sqrt{3}}{2}) \frac{2 \sqrt{3}}{3^{\frac{2}{2}}} = \frac{500}{\cos (30)}$

Rewrite the expression.

$(\frac{F \sqrt{3}}{2}) \frac{2 \sqrt{3}}{3^{1}} = \frac{500}{\cos (30)}$

Evaluate the exponent.

$(\frac{F \sqrt{3}}{2}) \frac{2 \sqrt{3}}{3} = \frac{500}{\cos (30)}$

Combine $(\frac{F \sqrt{3}}{2})$ and $\frac{2 \sqrt{3}}{3}$ .

$\frac{(\frac{F \sqrt{3}}{2}) (2 \sqrt{3})}{3} = \frac{500}{\cos (30)}$

Simplify the numerator.

Combine $\frac{F \sqrt{3}}{2}$ and $2$ .

$\frac{\frac{F \sqrt{3} \cdot 2}{2} \sqrt{3}}{3} = \frac{500}{\cos (30)}$

Combine $\frac{F \sqrt{3} \cdot 2}{2}$ and $\sqrt{3}$ .

$\frac{\frac{F \sqrt{3} \cdot 2 \sqrt{3}}{2}}{3} = \frac{500}{\cos (30)}$

Simplify the numerator.

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

$\frac{\frac{F \cdot 2 ({\sqrt{3}}^{1} \sqrt{3})}{2}}{3} = \frac{500}{\cos (30)}$

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

$\frac{\frac{F \cdot 2 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}{2}}{3} = \frac{500}{\cos (30)}$

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

$\frac{\frac{F \cdot 2 {\sqrt{3}}^{1 + 1}}{2}}{3} = \frac{500}{\cos (30)}$

Add $1$ and $1$ .

$\frac{\frac{F \cdot 2 {\sqrt{3}}^{2}}{2}}{3} = \frac{500}{\cos (30)}$

Simplify the numerator.

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{\frac{F \cdot 2 {(3^{\frac{1}{2}})}^{2}}{2}}{3} = \frac{500}{\cos (30)}$

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

$\frac{\frac{F \cdot 2 \cdot 3^{\frac{1}{2} \cdot 2}}{2}}{3} = \frac{500}{\cos (30)}$

Combine $\frac{1}{2}$ and $2$ .

$\frac{\frac{F \cdot 2 \cdot 3^{\frac{2}{2}}}{2}}{3} = \frac{500}{\cos (30)}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\frac{F \cdot 2 \cdot 3^{\frac{2}{2}}}{2}}{3} = \frac{500}{\cos (30)}$

Rewrite the expression.

$\frac{\frac{F \cdot 2 \cdot 3^{1}}{2}}{3} = \frac{500}{\cos (30)}$

Evaluate the exponent.

$\frac{\frac{F \cdot 2 \cdot 3}{2}}{3} = \frac{500}{\cos (30)}$

Multiply $3$ by $2$ .

$\frac{\frac{F \cdot 6}{2}}{3} = \frac{500}{\cos (30)}$

Reduce the expression by cancelling the common factors.

Reduce the expression $\frac{F \cdot 6}{2}$ by cancelling the common factors.

Factor $2$ out of $F \cdot 6$ .

$\frac{\frac{2 (F \cdot 3)}{2}}{3} = \frac{500}{\cos (30)}$

Factor $2$ out of $2$ .

$\frac{\frac{2 (F \cdot 3)}{2 (1)}}{3} = \frac{500}{\cos (30)}$

Cancel the common factor.

$\frac{\frac{2 (F \cdot 3)}{2 \cdot 1}}{3} = \frac{500}{\cos (30)}$

Rewrite the expression.

$\frac{\frac{F \cdot 3}{1}}{3} = \frac{500}{\cos (30)}$

Divide $F \cdot 3$ by $1$ .

$\frac{F \cdot 3}{3} = \frac{500}{\cos (30)}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{F \cdot 3}{3} = \frac{500}{\cos (30)}$

Divide $F$ by $1$ .

$F = \frac{500}{\cos (30)}$

Step 3

Simplify the right side.

Separate fractions.

$F = \frac{500}{1} \cdot \frac{1}{\cos (30)}$

Convert from $\frac{1}{\cos (30)}$ to $\sec (30)$ .

$F = \frac{500}{1} \sec (30)$

Divide $500$ by $1$ .

$F = 500 \sec (30)$

The exact value of $\sec (30)$ is $\frac{2}{\sqrt{3}}$ .

$F = 500 \frac{2}{\sqrt{3}}$

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$F = 500 (\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Combine and simplify the denominator.

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$F = 500 \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

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

$F = 500 \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

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

$F = 500 \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

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

$F = 500 \frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Add $1$ and $1$ .

$F = 500 \frac{2 \sqrt{3}}{{\sqrt{3}}^{2}}$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$F = 500 \frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

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

$F = 500 \frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$F = 500 \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$F = 500 \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}$

Rewrite the expression.

$F = 500 \frac{2 \sqrt{3}}{3^{1}}$

Evaluate the exponent.

$F = 500 \frac{2 \sqrt{3}}{3}$

Multiply $500 \frac{2 \sqrt{3}}{3}$ .

Combine $500$ and $\frac{2 \sqrt{3}}{3}$ .

$F = \frac{500 (2 \sqrt{3})}{3}$

Multiply $2$ by $500$ .

$F = \frac{1000 \sqrt{3}}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$F = \frac{1000 \sqrt{3}}{3}$

Decimal Form:

$F = 577.35026918 \dots$

Solve for F F*(cos(30))=500

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