We can find the area of a circle using the following formula:

\(\displaystyle A=s^2\)

In this equation the variable, \(\displaystyle s\), represents the length of a single side.

Substitute and solve.

\(\displaystyle A=5^2\)

\(\displaystyle A=25\) 

Since a square comprises four segments of the same length, the length of one side is equal to one fourth of the perimeter of the square, which is \(\displaystyle \frac{1}{4}P\). The area of the square is equal to the square of this sidelength, or

\(\displaystyle \left (\frac{1}{4}P \right )^{2} = \left (\frac{1}{4} \right )^{2} P^{2} = \frac{1}{16}P^{2}\).

The volume of a sphere can be calculated using the formula

\(\displaystyle V= \frac{4}{3} \pi r^{3}\)

Solving for \(\displaystyle r\):

Set \(\displaystyle V = 108 \pi\). Multiply both sides by \(\displaystyle \frac{3}{4}\):

\(\displaystyle \frac{3}{4} \cdot \frac{4}{3} \pi r^{3} = \frac{3}{4} \cdot 108 \pi\)

\(\displaystyle \pi r^{3} = 81\pi\)

Divide by \(\displaystyle \pi\):

\(\displaystyle \frac{\pi r^{3} }{\pi}= \frac{81\pi}{\pi}\)

\(\displaystyle r^{3} = 81\)

Take the cube root of both sides:

\(\displaystyle r = \sqrt[3]{81} = \sqrt[3]{27(3)} = \sqrt[3]{27} \cdot \sqrt[3]{3} = 3 \sqrt[3]{3}\)

Now substitute for \(\displaystyle r\) in the surface area formula:

\(\displaystyle A = 4 \pi r^{2}\)

\(\displaystyle A = 4 \pi (3 \sqrt[3]{3})^{2}= 4 \pi (3 )^{2}( \sqrt[3]{3})^{2} = 4 \pi (9) ( \sqrt[3]{3^{2}}) = 36 \pi \sqrt[3]{9}\),

the correct response.

One yard is equal to three feet, so convert 60 feet to yards by dividing by conversion factor 3:

\(\displaystyle 60 \textup{ ft} \div 3 \textup{ ft / yd}= 20 \textup{ yd}\)

Square this sidelength to get the area of the plot:

\(\displaystyle 20 \textup{ yd} \times 20 \textup{ yd} = 400 \textup{ yd} ^{2}\),

the correct response.

Divide the perimeter to get the length of one side of the square.

\(\displaystyle P = 12y + 4\)

\(\displaystyle s= \frac{P}{4}\)

\(\displaystyle s= \frac{12y+4}{4}\)

Divide each term by 4:

\(\displaystyle s= \frac{12y}{4}+ \frac{4}{4} = 3y+ 1\)

Square this sidelength to get the area of the square. The binomial can be squared by using the square of a binomial pattern:

\(\displaystyle A= s^{2}\)

\(\displaystyle =( 3y+ 1)^{2}\)

\(\displaystyle =( 3y )^{2}+ 2 (3y)(1)+ 1^{2}\)

\(\displaystyle = 3^{2}y ^{2}+ 2 (3 )(1)y+ 1^{2}\)

\(\displaystyle =9y ^{2}+ 6y+ 1\)

A cube with surface area 6 has six faces,each with area 1. As a result, each edge of the cube has length the square root of this, which is 1.

This is the diameter of the sphere inscribed in the cube, so the radius of the sphere is half this, or \(\displaystyle r= \frac{1}{2}\). Substitute this for \(\displaystyle r\) in the formula for the surface area of a sphere:

\(\displaystyle A = 4\pi r^{2} = 4\pi \left ( \frac{1}{2} \right )^{2} = 4\pi \left ( \frac{1}{4} \right ) = \pi\),

the correct choice.

The hypotenuse of a right triangle can be calculated using the Pythagorean Theorem. This theorem states that if we know the lengths of the two other legs of the triangle, then we can calculate the hypotenuse. It is written in the following way:

\(\displaystyle a^2+b^2=c^2\)

In this formula the legs are noted by the variables, \(\displaystyle a\) and \(\displaystyle b\). The variable \(\displaystyle c\) represents the hypotenuse.

Substitute and solve for the hypotenuse.

\(\displaystyle 3^2+5^2=c^2\)

Simplify.

\(\displaystyle 9+25=c^2\)

\(\displaystyle 34=c^2\)

Take the square root of both sides of the equation.

\(\displaystyle \sqrt{c^2}=\sqrt{34}\)

\(\displaystyle c=5.83\)

Step 1: Recall the Pythagorean theorem statement and formula.

Statement: For any right triangle, the sums of the squares of the shorter sides is equal to the square of the longest side.

Formula: In a right triangle \(\displaystyle ABC\), If \(\displaystyle a,b\) are the shorter sides and \(\displaystyle c\) is the longest side.. then,

\(\displaystyle a^2+b^2=c^2\)

Step 2: Plug in the values given to us in the problem....

\(\displaystyle (10)^2+(15)^2=c^2\)

Evaluate:

\(\displaystyle (10\times 10)+(15\times 15)=c^2\)

Simplify:

\(\displaystyle 100+225=c^2\)

Simplify:

\(\displaystyle 325=c^2\)

Take the square root...

\(\displaystyle \sqrt{325}=c\)

Step 3: Simplify the root...

\(\displaystyle \sqrt {325}=\sqrt {25\times13}=\sqrt{5\times 5\times 13}=5\sqrt {13}\)

The length of the hypotenuse in most simplified form is \(\displaystyle 5\sqrt{13}\) cm. 

In each choice, the two shortest sides of the triangle are 9 and 12, so the third side can be found by applying the Pythagorean Theorem. Set \(\displaystyle a= 9, b=12\) in the Pythagorean equation and solve for \(\displaystyle c\):

\(\displaystyle c^{2} = a^{2}+b^{2} = 9^{2}+12^{12} = 81 + 144= 225\)

Take the square root:

\(\displaystyle c = \sqrt{225}= 15\).

The correct choice is

\(\displaystyle 9, 12, 15\).

Reflecting a point

\(\displaystyle (x,y)\)

over the y-axis geometrically is the same as negating the x-coordinate of the ordered pair to obtain

\(\displaystyle (-x,y)\).

Thus, since our initial point was

\(\displaystyle (2,-2)\) 

and we want to reflect it over the y-axis, we obtain the reflection by negating the first term of the ordered pair to get

\(\displaystyle (-2,-2)\).