ISEE Upper Level Math : ISEE Upper Level (grades 9-12) Mathematics Achievement

Study concepts, example questions & explanations for ISEE Upper Level Math

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Example Questions

Example Question #21 : How To Find The Area Of A Circle

Find the area of a circle with a diameter of 16in.

Possible Answers:

\(\displaystyle 64\pi \text{ in}^2\)

\(\displaystyle 128\pi \text{ in}^2\)

\(\displaystyle 36\pi \text{ in}^2\)

\(\displaystyle 84\pi \text{ in}^2\)

\(\displaystyle 96\pi \text{ in}^2\)

Correct answer:

\(\displaystyle 64\pi \text{ in}^2\)

Explanation:

To find the area of a circle, we will use the following formula:

\(\displaystyle A = \pi r^2\)

where r is the radius of the circle. 

Now, we know the diameter is 16in. We also know the diameter is two times the radius. Therefore, the radius is 8in. So, we get

\(\displaystyle A = \pi \cdot (8\text{in})^2\)

\(\displaystyle A = \pi \cdot 64\text{in}^2\)

\(\displaystyle A = 64\pi \text{ in}^2\)

Example Question #22 : How To Find The Area Of A Circle

A circle has a diameter of 18cm. Find the area.

Possible Answers:

\(\displaystyle 36\pi \text{ cm}^2\)

\(\displaystyle 324\pi \text{ cm}^2\)

\(\displaystyle 81\pi \text{ cm}^2\)

\(\displaystyle 64\pi \text{ cm}^2\)

\(\displaystyle 124\pi \text{ cm}^2\)

Correct answer:

\(\displaystyle 81\pi \text{ cm}^2\)

Explanation:

To find the area of a circle, we will use the following formula:

\(\displaystyle A = \pi \cdot r^2\)

where r is the radius of the circle. 

 

Now, we know the diameter of the circle is 18cm.  We also know the diameter is two times the radius.  Therefore, the radius is 9cm.

Knowing this, we can substitute into the formula.  We get

\(\displaystyle A = \pi \cdot (9\text{cm})^2\)

\(\displaystyle A = \pi \cdot 81\text{cm}^2\)

\(\displaystyle A = 81\pi \text{ cm}^2\)

Example Question #3 : Perimeter

What is the circumference of a circle with a radius equal to \(\displaystyle 7\)?

Possible Answers:

\(\displaystyle 49\pi\)

\(\displaystyle 14\pi\)

\(\displaystyle 7\pi\)

\(\displaystyle 196\pi\)

Correct answer:

\(\displaystyle 14\pi\)

Explanation:

The circumference can be solved using the following equation:

\(\displaystyle C=2r\pi=2(7)\pi=14\pi\)

Example Question #1 : Circumference Of A Circle

The radius of a circle is \(\displaystyle t^2\). Give the circumference of the circle in terms of \(\displaystyle t\).

Possible Answers:

\(\displaystyle \pi t\)

\(\displaystyle 2\pi t^2\)

\(\displaystyle 2\pi t\)

\(\displaystyle \pi t^2\)

\(\displaystyle 4\pi t^2\)

Correct answer:

\(\displaystyle 2\pi t^2\)

Explanation:

The circumference can be calculated as \(\displaystyle Circumference =2\pi r\), where \(\displaystyle r\) is the radius of the circle.

\(\displaystyle Circumference =2\pi r=2\pi t^2\)

Example Question #61 : Circles

The area of a circle is \(\displaystyle 25t^2\). Give the circumference  of the circle in terms of \(\displaystyle t\).

Let \(\displaystyle \pi=3.14\).

Possible Answers:

\(\displaystyle 20t^2\)

\(\displaystyle 17.71 t^2\)

\(\displaystyle 15t\)

\(\displaystyle 17.71t\)

\(\displaystyle 20t\)

Correct answer:

\(\displaystyle 17.71t\)

Explanation:

The area of a circle can be calculated as \(\displaystyle Area=\pi r^2\), where \(\displaystyle r\)  is the radius of the circle. 

\(\displaystyle Area=\pi r^2\Rightarrow 25t^2=\pi r^2\Rightarrow r^2=\frac{25t^2}{\pi}\)

\(\displaystyle \Rightarrow r=\frac{5t}{\sqrt{\pi}}=\frac{5t}{\sqrt{3.14}}\approx 2.82t\)

The circumference can be calculated as \(\displaystyle Circumference =2\pi r\), where \(\displaystyle r\) is the radius of the circle.

\(\displaystyle Circumference =2\pi r\Rightarrow Circumference =2\times 3.14\times 2.82t=17.71t\)

 

 

 

Example Question #2 : Perimeter

What is the circumference of a circle with a radius of \(\displaystyle 8\)?

Possible Answers:

\(\displaystyle 28\pi\)

\(\displaystyle 16\pi^{2}\)

\(\displaystyle 16\pi\)

\(\displaystyle 16\)

\(\displaystyle 8\pi\)

Correct answer:

\(\displaystyle 16\pi\)

Explanation:

The circumference can be solved using the following equation:

\(\displaystyle C=2r\pi\)

Where \(\displaystyle r\) represents the radius. Therefore, when we substitute our radius in we get:

\(\displaystyle =2(8)\pi =16\pi\)

Example Question #1 : Know And Use The Formulas For The Area And Circumference Of A Circle: Ccss.Math.Content.7.G.B.4

The perimeter of a given rectangle is equal to the circumference of a given circle. The circle has radius \(\displaystyle 30\) inches; the width of the rectangle is \(\displaystyle 8 \pi\) inches. What is the length of the rectangle?

Possible Answers:

\(\displaystyle 14 \pi\) inches

\(\displaystyle 7\pi\) inches

\(\displaystyle 22 \pi\) inches

\(\displaystyle (30 - 8 \pi )\) inches

\(\displaystyle 44 \pi\) inches

Correct answer:

\(\displaystyle 22 \pi\) inches

Explanation:

The circumference of a circle with radius \(\displaystyle 30\) inches is 

\(\displaystyle C = 2 \pi r = 2 \pi \cdot 30 = 60 \pi\) inches.

The perimeter of the rectangle is therefore \(\displaystyle 60 \pi\) inches. To find its length, substitute \(\displaystyle W = 8\pi\) and \(\displaystyle P = 60\pi\) into the equation and solve for \(\displaystyle L\):

\(\displaystyle 2 L + 2W = P\)

\(\displaystyle 2 L + 2 \cdot 8 \pi = 60 \pi\)

\(\displaystyle 2 L + 16 \pi = 60 \pi\)

\(\displaystyle 2 L + 16 \pi - 16 \pi = 60 \pi - 16 \pi\)

\(\displaystyle 2 L =44 \pi\)

\(\displaystyle 2 L \div 2=44 \pi \div 2\)

\(\displaystyle L = 22\pi\) inches

Example Question #1 : How To Find Circumference

Sector

The area of the shaded region in the above diagram is \(\displaystyle 72 \pi\). Give the circumference of the circle.

Possible Answers:

\(\displaystyle 36 \pi\)

\(\displaystyle 18 \pi\)

\(\displaystyle 72 \pi\)

\(\displaystyle 8 \pi\)

Correct answer:

\(\displaystyle 36 \pi\)

Explanation:

The shaded region is an \(\displaystyle 80 ^{\circ }\) sector; this represents 

\(\displaystyle \frac{80}{360} = \frac{80 \div 40 }{360\div 40} = \frac{2}{9}\) 

of the circle. The area of the region is therefore

\(\displaystyle A = \frac{2}{9} \cdot \pi r^{2}\),

where \(\displaystyle r\) is the radius. Setting \(\displaystyle A = 72 \pi\) and solving for \(\displaystyle r\):

\(\displaystyle \frac{2}{9} \cdot \pi r^{2} = 72 \pi\)

\(\displaystyle \frac{9}{2 \pi} \cdot \frac{2}{9} \cdot \pi r^{2} =\frac{9}{2 \pi} \cdot 72 \pi\)

\(\displaystyle r^{2} =324\)

\(\displaystyle r = \sqrt{324 } =18\)

The circumference of a circle is its radius multiplied by \(\displaystyle 2 \pi\), or

\(\displaystyle C = 2 \pi r = 2 \pi \cdot 18 = 36 \pi\)

Example Question #191 : Isee Upper Level (Grades 9 12) Mathematics Achievement

A circle has a radius of 5 miles, what is its circumference?

 

Possible Answers:

\(\displaystyle 10\pi mi\)

\(\displaystyle 314 mi\)

\(\displaystyle 25 \pi mi\)

\(\displaystyle 125 mi\)

Correct answer:

\(\displaystyle 10\pi mi\)

Explanation:

A circle has a radius of 5 miles, what is its circumference?

To find circumference, use the following formula:

\(\displaystyle Circumference=2\pi r\)

Where r is our radius.

Plug in 5 miles to get our answer:

\(\displaystyle Circumference=2\pi (5mi)=10 \pi mi\)

Example Question #192 : Isee Upper Level (Grades 9 12) Mathematics Achievement

While mailing some very important letters, you decide to use your circular rubber stamp. If the stamp has a radius of 3.5 cm, what is the circumference of the stamping surface?

Possible Answers:

\(\displaystyle 7 \pi cm\)

\(\displaystyle 12.25 cm\)

Cannot be determined from the information provided.

\(\displaystyle 9.25cm\)

Correct answer:

\(\displaystyle 7 \pi cm\)

Explanation:

While mailing some very important letters, you decide to use your circular rubber stamp. If the stamp has a radius of 3.5 cm, what is the circumference of the stamping surface?

Find circumference via the following:

\(\displaystyle C=2 \pi r\)

Where r is our radius, which is 3.5 cm.

\(\displaystyle C=2(3.5cm) \pi =7\pi cm\)

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