Problem Solving

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HSPT Math › Problem Solving

Questions 1 - 10

Your backyard is wide and long, what is its area?

$60ft^{2}$

$16ft^{2}$

$50ft^{2}$

$10ft^{2}$

Explanation

The area of a rectangle is

So for this you just multiple those two values together to get,

$A=L\cdot w$

Remeber that the units of area are squared.

There are 500 students at the high school. There are only two menu options: chicken or fish. If 15% of students ordered fish, how many students ordered chicken?

$425\ students$

$75\ students$

$85\ students$

$450\ students$

$325\ students$

Explanation

If 15% of students ordered fish, then 85% of the students must have ordered chicken. Then multiply 85% or $.85\times 500 = 425$ .

A square with a side length of 4 inches is inscribed in a circle, as shown below. What is the area of the unshaded region inside of the circle, in square inches?

Act_math_01

8π - 16

4π-4

8π-4

2π-4

8π-8

Explanation

Using the Pythagorean Theorem, the diameter of the circle (also the diagonal of the square) can be found to be 4√2. Thus, the radius of the circle is half of the diameter, or 2√2. The area of the circle is then π(2√2)2, which equals 8π. Next, the area of the square must be subtracted from the entire circle, yielding an area of 8π-16 square inches.

What is the area of the figure below?

Explanation

To find the area of the figure above, we need to slip the figure into two rectangles.

3.5

Using our area formula, $A=l\times w$ , we can solve for the area of both of our rectangles

$A=11\times 3$ $A=3\times 3$

To find our final answer, we need to add the areas together.

Rectangle

Give the area of the above rectangle in square feet.

$216 \textrm{ ft}^{2}$

$532\textrm{ ft}^{2}$

$576 \textrm{ ft}^{2}$

$33 \textrm{ ft}^{2}$

$66 \textrm{ ft}^{2}$

Explanation

Since 1 yard = 3 feet, multiply each dimension by 3 to convert from yards to feet:

$8 \textrm{ yd} \times 3 \textrm{ ft/yd} = 24 \textrm{ ft}$

$3 \textrm{ yd} \times 3 \textrm{ ft/yd} = 9 \textrm{ ft}$

Use the area formula, substituting :

$A = 24 \times 9 = 216$ square feet

What is the area of the figure below?

Explanation

To find the area of the figure above, we need to slip the figure into two rectangles.

11.5

Using our area formula, $A=l\times w$ , we can solve for the area of both of our rectangles

$A=7\times 3$ $A=6\times 3$

To find our final answer, we need to add the areas together.

What angle is complement to $\frac{\pi}{6}$ ?

$\frac{\pi}{3}$

$\frac{\pi}{9}$

$\frac{\pi}{4}$

$\frac{\pi}{6}$

$\frac{2\pi}{3}$

Explanation

The complement to an angle is ninety degrees subtract the angle since two angles must add up to 90. In this case, since we are given the angle in radians, we are subtracting from $\frac{\pi}{2}$ instead to find the complement. The conversion between radians and degrees is: $\pi \textup{ radians } = 180 \textup{ degrees}$

$\frac{\pi}{2}-\frac{\pi}{6}$

Reconvert the fractions to the least common denominator.

$\frac{\pi}{2}-\frac{\pi}{6} = \frac{3\pi}{6}-\frac{\pi}{6} = \frac{2\pi}{6}$

Reduce the fraction.

$\frac{2\pi}{6} = \frac{\pi}{3}$

The ratio of the perimeter of one square to that of another square is . What is the ratio of the area of the first square to that of the second square?

Explanation

For the sake of simplicity, we will assume that the second square has sidelength 1; Then its perimeter is $4 \times 1 = 4$ , and its area is $1^{2} = 1$ .

The perimeter of the first square is $\frac{7}{4} \times 4 = 7$ , and its sidelength is $7 \div 4 = \frac{7}{4}$ . The area of this square is therefore $\left (\frac{7}{4} \right )^{2} =\frac{49}{16}$ .

The ratio of the areas is therefore $\frac{49}{16} : 1 = 49:16$ .

On a particular November day, it is $110^\circ F$ and sunny in Tucson, Arizona. On the eastern side of the United States, it is $-15^\circ F$ and sleeting in New Castle, Pennsylvania. How many degrees warmer is it in Tucson than New Castle?

$125^\circ F$

$152^\circ F$

$150^\circ F$

$115^\circ F$

$80^\circ F$

Explanation

There are several ways that we could solve this problem. First, we can say that the temperature in New Castle is $15^\circ F$ below zero and in Tucson it is $110^\circ F$ above zero; therefore we can say: