10. evaluate the expression for the area of a regular polygon ( a = \frac{1}{2}ap ) for ( a = 2sqrt{3} ) and ( p = 60 ).
11. solve the equation for the area of a circle for the variable ( r ).
12. solve the proportion diagram ( \frac{3x}{5} = \frac{7}{14} ) (approximate ocr of the proportion).
13. a ladder makes a ( 50^circ ) angle with the ground and reaches 20 feet up the side of a building. find to the nearest tenth how far the base of the ladder is away from the base of the building.

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Accepted Answer

### Question 10
# Explanation:
## Step1: Substitute the values of $a$ and $p$ into the formula.
The formula for the area of a regular polygon is $A=\frac{1}{2}ap$. We are given $a = 2\sqrt{3}$ and $p=60$. Substitute these values into the formula: $A=\frac{1}{2}\times(2\sqrt{3})\times60$
## Step2: Simplify the expression.
First, $\frac{1}{2}\times2 = 1$, so the expression becomes $1\times\sqrt{3}\times60=60\sqrt{3}$. If we calculate the numerical value, $60\sqrt{3}\approx60\times1.732 = 103.92$
# Answer: $60\sqrt{3}$ (or approximately $103.9$ if a decimal approximation is needed)

### Question 11
# Explanation:
## Step1: Recall the formula for the area of a circle.
The formula for the area of a circle is $A=\pi r^{2}$, where $A$ is the area and $r$ is the radius.
## Step2: Solve for $r$.
First, divide both sides of the equation by $\pi$: $\frac{A}{\pi}=r^{2}$. Then, take the square root of both sides. Since the radius $r>0$ (length cannot be negative), we have $r = \sqrt{\frac{A}{\pi}}$ (or $r=\frac{\sqrt{A\pi}}{\pi}$ after rationalizing the denominator)
# Answer: $r=\sqrt{\frac{A}{\pi}}$ (or equivalent forms like $r = \frac{\sqrt{A\pi}}{\pi}$)

### Question 12
(Assuming the proportion is $\frac{3x}{5}=\frac{7}{14}$ or a similar form, let's assume the proportion is $\frac{3x}{5}=\frac{7}{14}$ for correction. If the original proportion is $\frac{3x}{5}=\frac{7}{14}$)
# Explanation:
## Step1: Cross - multiply.
For a proportion $\frac{a}{b}=\frac{c}{d}$, we have $a\times d=b\times c$. So for $\frac{3x}{5}=\frac{7}{14}$, cross - multiplying gives $3x\times14 = 5\times7$
## Step2: Simplify and solve for $x$.
Simplify the right - hand side: $5\times7 = 35$. The left - hand side: $3x\times14=42x$. So we have the equation $42x = 35$. Divide both sides by 42: $x=\frac{35}{42}=\frac{5}{6}$
(If the proportion is $\frac{3x}{5}=\frac{7}{14}$ as assumed)
# Answer: $x = \frac{5}{6}$

### Question 13
# Explanation:
## Step1: Identify the trigonometric relationship.
We can model this situation as a right triangle, where the height on the building is the opposite side ($opp = 20$ feet) to the angle of $50^{\circ}$ (with the ground), and the distance from the base of the ladder to the base of the building is the adjacent side ($adj$) to the $50^{\circ}$ angle. We know that $\tan\theta=\frac{opp}{adj}$, so $\tan(50^{\circ})=\frac{20}{adj}$
## Step2: Solve for $adj$.
Rearrange the formula to solve for $adj$: $adj=\frac{20}{\tan(50^{\circ})}$. We know that $\tan(50^{\circ})\approx1.1918$. So $adj=\frac{20}{1.1918}\approx16.8$
# Answer: Approximately $16.8$ feet

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QUESTION IMAGE

Question

Explanation:

Question 10

Step1: Substitute the values of \(a\) and \(p\) into the formula.

Step2: Simplify the expression.

Step1: Recall the formula for the area of a circle.

Step2: Solve for \(r\).

Step1: Cross - multiply.

Step2: Simplify and solve for \(x\).

Answer:

Question 11