HiSET: Math : HiSet: High School Equivalency Test: Math

Study concepts, example questions & explanations for HiSET: Math

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Example Questions

Example Question #91 : Hi Set: High School Equivalency Test: Math

Find the area of a square with the following side length:

\(\displaystyle s=5\)

Possible Answers:

\(\displaystyle 25\)

\(\displaystyle 10\)

\(\displaystyle 15\)

\(\displaystyle 20\)

\(\displaystyle 5\)

Correct answer:

\(\displaystyle 25\)

Explanation:

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.

\(\displaystyle A=5^2\)

\(\displaystyle A=25\) 

Example Question #32 : Measurement And Geometry

The perimeter of a square is \(\displaystyle P\). In terms of \(\displaystyle P\), give the area of the square.

Possible Answers:

\(\displaystyle \frac{1}{16}P^{2}\)

\(\displaystyle \frac{1}{4} \sqrt{P}\)

\(\displaystyle 4P\)

\(\displaystyle \frac{1}{16} \sqrt{P}\)

\(\displaystyle \frac{1}{4}P^{2}\)

Correct answer:

\(\displaystyle \frac{1}{16}P^{2}\)

Explanation:

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

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

Example Question #92 : Hi Set: High School Equivalency Test: Math

The volume of a sphere is equal to \(\displaystyle 108 \pi\). Give the surface area of the sphere.

Possible Answers:

\(\displaystyle 12 \pi \sqrt[3]{9}\)

None of the other choices gives the correct response.

\(\displaystyle 36 \pi \sqrt[3]{3}\)

\(\displaystyle 12 \pi \sqrt[3]{3}\)

\(\displaystyle 36 \pi \sqrt[3]{9}\)

Correct answer:

\(\displaystyle 36 \pi \sqrt[3]{9}\)

Explanation:

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.

Example Question #93 : Hi Set: High School Equivalency Test: Math

Express the area of a square plot of land 60 feet in sidelength in square yards.

Possible Answers:

3,600 square yards

200 square yards

600 square yards

400 square yards

600 square yards

Correct answer:

400 square yards

Explanation:

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.

Example Question #27 : Properties Of Polygons And Circles

A square has perimeter \(\displaystyle 12y + 4\). Give its area in terms of \(\displaystyle y\).

Possible Answers:

\(\displaystyle y ^{2}+ 8y+ 16\)

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

\(\displaystyle 9y ^{2}+ 24y+ 16\)

\(\displaystyle 9y ^{2}+ 12y+ 4\)

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

Correct answer:

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

Explanation:

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:

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

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

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

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

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

 

Example Question #34 : Measurement And Geometry

A cube has surface area 6. Give the surface area of the sphere that is inscribed inside it.

Possible Answers:

\(\displaystyle \frac{1}{6} \pi\)

\(\displaystyle \frac{1}{3} \pi\)

\(\displaystyle \pi\)

\(\displaystyle \frac{2}{3} \pi\)

\(\displaystyle \frac{1}{2} \pi\)

Correct answer:

\(\displaystyle \pi\)

Explanation:

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.

Example Question #94 : Hi Set: High School Equivalency Test: Math

Find the length of the hypotenuse of a right triangle whose legs are the following lengths:

\(\displaystyle a=3 \textup{ and }b=5\)

Possible Answers:

\(\displaystyle 8.53\)

\(\displaystyle 7\)

\(\displaystyle 3.85\)

\(\displaystyle 4\)

\(\displaystyle 5.83\)

Correct answer:

\(\displaystyle 5.83\)

Explanation:

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.

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

\(\displaystyle c=5.83\)

Example Question #91 : Hi Set: High School Equivalency Test: Math

If the two legs of a right triangle are \(\displaystyle 10\) cm and \(\displaystyle 15\) cm, what is the length of the hypotenuse. Answer must be in SIMPLIFIED form (or lowest terms).

Possible Answers:

\(\displaystyle 18.03\) cm

\(\displaystyle 5\sqrt{11}\) cm

\(\displaystyle 5\sqrt{13}\) cm

\(\displaystyle 5\sqrt{17}\) cm

Correct answer:

\(\displaystyle 5\sqrt{13}\) cm

Explanation:

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. 

Example Question #3 : Understand Right Triangles

Which of the following could be the lengths of the sides of a right triangle?

Possible Answers:

\(\displaystyle 9, 12, 14\)

\(\displaystyle 9, 12, 17\)

\(\displaystyle 9, 12, 16\)

\(\displaystyle 9, 12, 15\)

\(\displaystyle 9, 12, 18\)

Correct answer:

\(\displaystyle 9, 12, 15\)

Explanation:

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

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

 

Example Question #92 : Hi Set: High School Equivalency Test: Math

What is the result of reflecting the point \(\displaystyle (2,-2)\) over the y-axis in the coordinate plane?

Possible Answers:

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

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

\(\displaystyle (2,2)\)

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

\(\displaystyle (-\frac{1}{2},2)\)

Correct answer:

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

Explanation:

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

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

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