Problem Solving

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Questions 1 - 10

Spinner target

Above is a target. The radius of the smaller quarter-circles is half that of the larger quarter-circles.

A blindfolded man throws a dart at the above target. Assuming the dart hits the target, and that no skill is involved, give the odds against the dart landing in the yellow region.

39 to 1

13 to 1

14 to 1

40 to 1

Explanation

Call the radius of one of the smaller quarter-circles 1 (the reasoning is independent of the actual radius). Then the area of each quarter-circle is

$\frac{1}{4} \cdot \pi r ^{2} = \frac{1}{4} \cdot \pi \cdot 1 ^{2} = \frac{1}{4} \pi$ .

Each of the four wedges of one such quarter-circle has area

$\frac{1}{4} \cdot \frac{1}{4} \pi = \frac{\pi}{16}$ .

The yellow region is one such wedge.

The radius of each of the larger quarter-circles is 2, so the area of each is $\frac{1}{4} \cdot \pi r ^{2} = \frac{1}{4} \cdot \pi \cdot 2 ^{2} = \pi$

The total area of the target is

$\frac{1}{4}\pi + \frac{1}{4} \pi + \pi + \pi = \frac{5 \pi}{2}$

Therefore, the yellow wedge is

$\frac{\pi}{16} \div \frac{5 \pi}{2} = \frac{\pi}{16} \cdot \frac{2}{5 \pi}= \frac{1}{8} \cdot \frac{1}{5 }= \frac{1}{40}$

of the target, and the odds against the dart landing in that region are 39 to 1.

Find the area of a square if the length is .

Explanation

The area of a square is:

Substitute the length and simplify.

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.

6.5

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

$A=7\times 12$ $A=7\times 5$

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

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.

11.5

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

$A=8\times 4$ $A=7\times 4$

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

Find the area of a square if the length is .

Explanation

The area of a square is:

Substitute the length and simplify.

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.

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.

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.

12.5

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

$A=6\times 8$ $A=7\times 5$

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: $\pi \textup{ radians } = 180 \textup{ degrees}$