ISEE Upper Level Quantitative : ISEE Upper Level (grades 9-12) Quantitative Reasoning

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

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Example Questions

Example Question #3 : Pyramids

A pyramid with a square base and a cone have the same height and the same volume. Which is the greater quantity?

(A) The perimeter of the base of the pyramid

(B) The circumference of the base of the cone

Possible Answers:

It is impossible to determine which is greater from the information given

(A) and (B) are equal

(B) is greater

(A) is greater

Correct answer:

(A) is greater

Explanation:

The volume of a pyramid or a cone with height \(\displaystyle h\) and base of area \(\displaystyle B\) is 

\(\displaystyle V = \frac{1}{3} Bh\)

so in both cases, the area of the base is

\(\displaystyle \frac{1}{3} Bh = V\)

\(\displaystyle \frac{3}{h} \cdot \frac{1}{3} Bh =\frac{3}{h} \cdot V\)

\(\displaystyle B = \frac{3V}{h}\)

Since the pyramid and the cone have the same volume and height, their bases has the same area \(\displaystyle B\).

The length of one side of the square base of the pyramid is the square root of this, or \(\displaystyle \sqrt{B}\), and the perimeter is four times this, or \(\displaystyle P = 4 \sqrt{B}\).

The radius and the area of the base of the cone are related as follows:

\(\displaystyle \pi r ^{2} = B\)

\(\displaystyle \sqrt{\pi r ^{2} }= \sqrt{B}\)

\(\displaystyle \sqrt{\pi } \cdot r = \sqrt{B}\)

Multiply both sides by \(\displaystyle 2 \sqrt{\pi}\) to get:

\(\displaystyle 2 \sqrt{\pi} \cdot \sqrt{\pi } \cdot r =2 \sqrt{\pi} \cdot \sqrt{B}\)

\(\displaystyle 2\pi r =2 \sqrt{\pi} \cdot \sqrt{B}\)

\(\displaystyle C =2 \sqrt{\pi} \cdot \sqrt{B}\)

\(\displaystyle 2 \sqrt{\pi} < 2 \sqrt{4} = 2 \times 2 = 4\), so 

\(\displaystyle 2 \sqrt{\pi} \cdot \sqrt{B} < 4 \sqrt{B}\), and

\(\displaystyle C < P\)

The perimeter of the base of the pyramid, which is (A), is greater than the circumference of the base of the cone.

Example Question #1 : Cubes

Which is the greater quantity?

(a) The volume of a cube with surface area  inches

(b) The volume of a cube with diagonal  inches

Possible Answers:

(b) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

(a) is greater.

Correct answer:

(b) is greater.

Explanation:

The cube with the greater sidelength has the greater volume, so we need only calculate and compare sidelengths.

(a) \(\displaystyle A=6s^{2}\), so the sidelength of the first cube can be found as follows:

\(\displaystyle A=6s^{2}\)

\(\displaystyle 6s^{2}= 864\)

\(\displaystyle 6s^{2} \div 6= 864 \div 6\)

\(\displaystyle s^{2} = 144\)

\(\displaystyle s = \sqrt{144 }= 12\) inches

(b) \(\displaystyle d^{2} = s^{2}+ s^{2}+ s^{2} = 3 s^{2}\) by an extension of the Pythagorean Theorem, so the sidelength of the second cube can be found as follows:

\(\displaystyle 3 s^{2}= d^{2}\)

\(\displaystyle 3 s^{2}= 21^{2}= 441\)

\(\displaystyle 3 s^{2}\div 3= 441 \div 3\)

\(\displaystyle s^{2}= 147\)

\(\displaystyle s=\sqrt{ 147}\)

Since \(\displaystyle 147 > 144\)\(\displaystyle \sqrt{147 }> \sqrt{144}\). The second cube has the greater sidelength and, subsequently, the greater volume. This makes (b) greater.

Example Question #1 : How To Find The Volume Of A Cube

Cube 2 has twice the sidelength of Cube 1; Cube 3 has twice the sidelength of Cube 2; Cube 4 has twice the sidelength of Cube 3.

Which is the greater quantity?

(a) The mean of the volumes of Cube 1 and Cube 4

(b) The mean of the volumes of Cube 2 and Cube 3

Possible Answers:

It cannot be determined from the information given.

(a) is greater.

(b) is greater.

(a) and (b) are equal.

Correct answer:

(a) is greater.

Explanation:

The sidelengths of Cubes 1, 2, 3, and 4 can be given values \(\displaystyle s, 2s, 4s, 8s\), respectively.

Then the volumes of the cubes are as follows:

Cube 1: \(\displaystyle V= s^{3}\)

Cube 2: \(\displaystyle V= (2s)^{3} = 8s^{3}\)

Cube 3: \(\displaystyle V= (4s)^{3} = 64s^{3}\)

Cube 4: \(\displaystyle V= (8s)^{3} = 512s^{3}\)

In both answer choices ask for a mean, so we can determine which answer (mean) is greater simply by comparing the sums of volumes.

(a) The sum of the volumes of Cubes 1 and 4 is \(\displaystyle s^{3}+ 512^{3} = 513s^{3}\).

(b) The sum of the volumes of Cubes 2 and 3 is \(\displaystyle 8s^{3}+ 64^{3} = 72s^{3}\).

Regardless of \(\displaystyle s\), the sum of the volumes of Cubes 1 and 4 is greater, and therefore, so is their mean.

Example Question #3 : Cubes

What is the volume of a cube with side length \(\displaystyle 7.236\) \(\displaystyle in\)? Round your answer to the nearest hundredth.

Possible Answers:

\(\displaystyle 314.16\) \(\displaystyle in^3\)

\(\displaystyle 612.32\) \(\displaystyle in^3\)

\(\displaystyle 378.87\) \(\displaystyle in^3\)

\(\displaystyle 123.41\) \(\displaystyle in^3\)

\(\displaystyle 452.31\) \(\displaystyle in^3\)

Correct answer:

\(\displaystyle 378.87\) \(\displaystyle in^3\)

Explanation:

This question is relatively straightforward. The equation for the volume of a cube is:

\(\displaystyle V = s^3\)

(It is like doing the area of a square, then adding another dimension!)

Now, for our data, we merely need to "plug and chug:"

\(\displaystyle v=7.236^3 =378.874760256\)

Example Question #1 : Cubes

What is the volume of a cube on which one face has a diagonal of \(\displaystyle 2\) \(\displaystyle in\)?

Possible Answers:

\(\displaystyle 12\) \(\displaystyle in^3\)

\(\displaystyle \frac{1}{2\sqrt{2} }\) \(\displaystyle in^3\)

\(\displaystyle 2\sqrt{2}\)  \(\displaystyle in^3\)

\(\displaystyle 2\) \(\displaystyle in^3\)

\(\displaystyle 12\sqrt{2}\) \(\displaystyle in^3\)

Correct answer:

\(\displaystyle 2\sqrt{2}\)  \(\displaystyle in^3\)

Explanation:

One of the faces of the cube could be drawn like this:

 

Squarediagonal-2

Notice that this makes a \(\displaystyle 45-45-90\) triangle.

This means that we can create a proportion for the sides. On the standard triangle, the non-hypotenuse sides are both \(\displaystyle 1\), and the hypotenuse is \(\displaystyle \sqrt{2}\).  This will allow us to make the proportion:

\(\displaystyle \frac{1}{\sqrt{2}} = \frac{x}{2}\)

Multiplying both sides by \(\displaystyle 2\), you get:

\(\displaystyle x=\frac{2}{\sqrt{2}}\)

Recall that the formula for the volume of a cube is:

\(\displaystyle V = s^3\)

Therefore, we can compute the volume using the side found above:

\(\displaystyle V = (\frac{2}{\sqrt{2}})^3=\frac{8}{(\sqrt{2})^3}=\frac{8}{2\sqrt{2}}=\frac{4}{\sqrt2}\)

Now, rationalize the denominator:

\(\displaystyle \frac{4}{\sqrt2} * \frac{\sqrt{2}}{\sqrt{2}} = \frac{4\sqrt{2}}{2}= 2\sqrt{2}\)

Example Question #4 : Cubes

The volume of a cube is 343 cubic inches. Give its surface area.

Possible Answers:

\(\displaystyle 343 \; \textrm{in}^{2}\)

\(\displaystyle 294 \; \textrm{in}^{2}\)

\(\displaystyle 196 \; \textrm{in}^{2}\)

\(\displaystyle 242\; \textrm{in}^{2}\)

\(\displaystyle 392 \; \textrm{in}^{2}\)

Correct answer:

\(\displaystyle 294 \; \textrm{in}^{2}\)

Explanation:

The volume of a cube is defined by the formula

\(\displaystyle V=s^{3}\)

where \(\displaystyle s\) is the length of one side.

If \(\displaystyle V=343\), then 

\(\displaystyle s^{3} = 343\)

and 

\(\displaystyle s = \sqrt[3]{343} = 7\)

So one side measures 7 inches. 

The surface area of a cube is defined by the formula

\(\displaystyle A = 6s^{2}\) , so

\(\displaystyle A = 6 \cdot 7^{2} = 294\)

The surface area is 294 square inches.

Example Question #5 : Cubes

What is the surface area of a cube with side length \(\displaystyle 5.7\) \(\displaystyle in\)?

Possible Answers:

\(\displaystyle 91.18\) \(\displaystyle in^2\)

\(\displaystyle 173.47\) \(\displaystyle in^2\)

\(\displaystyle 32.49\) \(\displaystyle in^2\)

\(\displaystyle 185.193\) \(\displaystyle in^2\)

\(\displaystyle 194.94\) \(\displaystyle in^2\)

Correct answer:

\(\displaystyle 194.94\) \(\displaystyle in^2\)

Explanation:

Recall that the formula for the surface area of a cube is:

\(\displaystyle SA = 6 * s^2\), where \(\displaystyle s\) is the length of a side of the cube. This equation is easy to memorize because it is merely a multiplication of a single side (\(\displaystyle s^2\)) by \(\displaystyle 6\) because a cube has \(\displaystyle 6\) equal sides.

For our data, we know that \(\displaystyle s = 5.7\); therefore, our equation is:

\(\displaystyle SA = 6 * 5.7^2 = 6 * 32.49 = 194.94\)

Example Question #6 : Cubes

What is the surface area of a cube with a volume \(\displaystyle 512\) \(\displaystyle in^3\)?

Possible Answers:

\(\displaystyle 193.44\) \(\displaystyle in^2\)

\(\displaystyle 384\) \(\displaystyle in^2\)

\(\displaystyle 192\) \(\displaystyle in^2\)

\(\displaystyle 321\) \(\displaystyle in^2\)

\(\displaystyle 85.33\) \(\displaystyle in^2\)

Correct answer:

\(\displaystyle 384\) \(\displaystyle in^2\)

Explanation:

To solve this, first calculate the side length based on the volume given. Recall that the equation for the volume of a cube is:

\(\displaystyle V = s^3\), where \(\displaystyle s\) is the side length.

For our data, this gives us:

\(\displaystyle s^3 = 512\)

Taking the cube-root of both sides, we get:

\(\displaystyle s = 8\)

Now, use the surface area formula to compute the total surface area:

\(\displaystyle SA = 6 * s^2\), where \(\displaystyle s\) is the length of a side of the cube. This equation is easy to memorize because it is merely a multiplication of a single side (\(\displaystyle s^2\)) by \(\displaystyle 6\) because a cube has \(\displaystyle 6\) equal sides.

For our data, this gives us:

\(\displaystyle SA = 6*8^2 = 6*64=384\)

Example Question #1 : How To Find The Surface Area Of A Cube

What is the surface area of a cube with a volume \(\displaystyle 274.625\) \(\displaystyle in^3\)?

Possible Answers:

\(\displaystyle 126.75\) \(\displaystyle in^2\)

\(\displaystyle 631.13\) \(\displaystyle in^2\)

\(\displaystyle 253.5\) \(\displaystyle in^2\)

\(\displaystyle 549.25\) \(\displaystyle in^2\)

\(\displaystyle 344.13\) \(\displaystyle in^2\)

Correct answer:

\(\displaystyle 253.5\) \(\displaystyle in^2\)

Explanation:

To solve this, first calculate the side length based on the volume given. Recall that the equation for the volume of a cube is:

\(\displaystyle V = s^3\), where \(\displaystyle s\) is the side length.

For our data, this gives us:

\(\displaystyle s^3 = 274.625\)

Taking the cube-root of both sides, we get:

\(\displaystyle s = 6.5\)

(You will need to use a calculator for this.  If your calculator gives you something like \(\displaystyle 6.599999999999\) . . . it is okay to round. This is just the nature of taking roots!).

Now, use the surface area formula to compute the total surface area:

\(\displaystyle SA = 6 * s^2\), where \(\displaystyle s\) is the length of a side of the cube. This equation is easy to memorize because it is merely a multiplication of a single side (\(\displaystyle s^2\)) by \(\displaystyle 6\) because a cube has \(\displaystyle 6\) equal sides.

For our data, this gives us:

\(\displaystyle SA = 6*6.5^2 = 6*42.25=253.5\)

Example Question #8 : Cubes

What is the surface area for a cube with a diagonal length of \(\displaystyle 3\sqrt{3}\) \(\displaystyle in\)?

Possible Answers:

\(\displaystyle 54\)

\(\displaystyle 28\sqrt{3}\)

\(\displaystyle 15\sqrt{6}\)

\(\displaystyle 3\sqrt{3}\)

\(\displaystyle 27\)

Correct answer:

\(\displaystyle 54\)

Explanation:

Now, this could look like a difficult problem; however, think of the equation for finding the length of the diagonal of a cube. It is like the Pythagorean Theorem, just adding an additional dimension:

\(\displaystyle D = \sqrt{3s^2}\)

(It is very easy, because the three lengths are all the same: \(\displaystyle s\)).

So, we know this, then:

\(\displaystyle 3\sqrt{3}=\sqrt{3s^2}\)

To solve, you can factor out an \(\displaystyle s\) from the root on the right side of the equation:

\(\displaystyle 3\sqrt{3}=s\sqrt{3}\)

Just by looking at this, you can tell that the answer is:

\(\displaystyle s=3\)

Now, use the surface area formula to compute the total surface area:

\(\displaystyle SA = 6 * s^2\), where \(\displaystyle s\) is the length of a side of the cube. This equation is easy to memorize because it is merely a multiplication of a single side (\(\displaystyle s^2\)) by \(\displaystyle 6\) because a cube has \(\displaystyle 6\) equal sides.

For our data, this is:

\(\displaystyle SA = 6 * 3^2 = 6*9=54\)

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