SAT II Math II : SAT Subject Test in Math II

Study concepts, example questions & explanations for SAT II Math II

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Example Questions

Example Question #2 : Area

You have a right triangle with a hypotenuse of 13 inches and a leg of 5 inches, what is the area of the triangle?

Possible Answers:

\(\displaystyle 120in^2\)

\(\displaystyle 56in^2\)

\(\displaystyle 12in^2\)

\(\displaystyle 30 in^2\)

Correct answer:

\(\displaystyle 30 in^2\)

Explanation:

You have a right triangle with a hypotenuse of 13 inches and a leg of 5 inches, what is the area of the triangle?

So find the area of a triangle, we need the following formula:

\(\displaystyle A_{tri}=\frac{1}{2}bh\)

However, we only know one leg, so we only know b or h.

To find the other leg, we can either use Pythagorean Theorem, or recognize that this is a 5-12-13 triangle. Meaning, our final leg is 12 inches long.

To prove this:

\(\displaystyle (13in)^2=(5in)^2+(xin)^2\)

\(\displaystyle 169in^2-25in^2=144in^2=(xin)^2\)

\(\displaystyle 144in^2=x^2in\)

\(\displaystyle x=\sqrt{144in^2}=12in\)

Now, we know both legs, let's just plug in and solve for area:

\(\displaystyle A_{tri}=\frac{1}{2}(5in)(12in)=6in*5in=30in^2\)

Example Question #3 : Area

You have a rectangular-shaped rug which you want to put in your living room. If the rug is 12.5 feet long and 18 inches wide, what is the area of the rug?

Possible Answers:

\(\displaystyle 18.75ft^2\)

\(\displaystyle 18.75ft\)

\(\displaystyle 3.75ft^2\)

\(\displaystyle 15ft^2\)

Correct answer:

\(\displaystyle 18.75ft^2\)

Explanation:

You have a rectangular-shaped rug which you want to put in your living room. If the rug is 12.5 feet long and 18 inches wide, what is the area of the rug?

To begin, we need to realize two things. 

1) Our given measurements are not in equivalent units, so we need to convert one of them before doing any solving.

2) The area of a rectangle is given by: \(\displaystyle A=l*w\)

Now, let's convert 18 inches to feet, because it seems easier than 12.5 feet to inches:

\(\displaystyle 18inches*\frac{1ft}{12in}=1.5ft\)

Now, using what we know from 2) we can find our answer

\(\displaystyle A=12.5ft*1.5ft=18.75ft^2\)

Example Question #1 : Geometry

Give the area of \(\displaystyle \bigtriangleup ABC\) to the nearest whole square unit, where:

\(\displaystyle AB = 34\)

\(\displaystyle AC = 27\)

\(\displaystyle m \angle A = 124^{\circ }\)

Possible Answers:

\(\displaystyle 459\)

\(\displaystyle 257\)

\(\displaystyle 513\)

\(\displaystyle 761\)

\(\displaystyle 381\)

Correct answer:

\(\displaystyle 381\)

Explanation:

The area of a triangle with two sides of lengths \(\displaystyle a\) and \(\displaystyle b\) and included angle of measure \(\displaystyle \gamma\) can be calculated using the formula

\(\displaystyle A = \frac{1}{2} ab \sin \gamma\).

Setting \(\displaystyle a = AB = 34, b= AC = 27 , \gamma = m \angle A = 124^{\circ }\) and evaluating \(\displaystyle A\):

\(\displaystyle A = \frac{1}{2} (34 )(27)\sin 124 ^{\circ }\)

\(\displaystyle \approx 459 ( 0.8290 )\)

\(\displaystyle \approx 381\)

Example Question #1 : Geometry

Give the area of \(\displaystyle \bigtriangleup ABC\) to the nearest whole square unit, where:

\(\displaystyle AB = 20\)

\(\displaystyle BC = 26\)

\(\displaystyle AC = 32\)

Possible Answers:

\(\displaystyle 129\)

\(\displaystyle 260\)

\(\displaystyle 260\)

\(\displaystyle 416\)

\(\displaystyle 320\)

Correct answer:

\(\displaystyle 260\)

Explanation:

The area of a triangle, given its three sidelengths, can be calculated using Heron's formula:

\(\displaystyle A = \sqrt{s(s-a)(s-b)(s-c)}\),

where \(\displaystyle a\)\(\displaystyle b\), and \(\displaystyle c\) are the lengths of the sides, and \(\displaystyle s = \frac{a+b+ c}{2}\).

Setting \(\displaystyle a = AB = 20\)\(\displaystyle b = BC = 26\), and \(\displaystyle c = AC = 32\), evaluate \(\displaystyle s\):

\(\displaystyle s = \frac{20+26+32}{2} = \frac{78}{2} = 39\)

and, substituting in Heron's formula:

\(\displaystyle A = \sqrt{39(39-20)(39-26)(39-32)}\)

\(\displaystyle = \sqrt{39(19)(13)(7)}\)

\(\displaystyle = \sqrt{67,431 }\)

\(\displaystyle \approx 259.7\)

To the nearest whole, this is 260.

Example Question #311 : Sat Subject Test In Math Ii

Find the area of a triangle with a base length of \(\displaystyle x^2\) and a height of \(\displaystyle \frac{1}{2}x\).

Possible Answers:

\(\displaystyle \frac{1}{2}x^3\)

\(\displaystyle x^3\)

\(\displaystyle x^2\)

\(\displaystyle \frac{1}{4}x^3\)

\(\displaystyle 4x^2\)

Correct answer:

\(\displaystyle \frac{1}{4}x^3\)

Explanation:

Write the formula for the area of a triangle.

\(\displaystyle A=\frac{BH}{2}\)

Substitute the dimensions.

\(\displaystyle A = \frac{x^2\cdot \frac{1}{2}x}{2} = \frac{1}{2}(x^2\cdot \frac{1}{2}x)\)

The answer is:  \(\displaystyle \frac{1}{4}x^3\)

Example Question #11 : Geometry

Find the area of a circle with a radius of \(\displaystyle \frac{3}{2}\).

Possible Answers:

\(\displaystyle 3\pi\)

\(\displaystyle 3\)

\(\displaystyle 9\pi\)

\(\displaystyle \frac{9}{2}\pi\)

\(\displaystyle \frac{9}{4}\pi\)

Correct answer:

\(\displaystyle \frac{9}{4}\pi\)

Explanation:

Write the formula for the area of a circle.

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

Substitute the radius into the equation.

\(\displaystyle A=\pi (\frac{3}{2})^2 = \frac{9}{4} \pi\)

The area is:  \(\displaystyle \frac{9}{4}\pi\)

Example Question #12 : Geometry

Determine the area of a rectangle if the length is \(\displaystyle x+3\) and the height is \(\displaystyle \frac{1}{4}(x+3)\).

Possible Answers:

\(\displaystyle \frac{1}{4}x^2 + 3x + \frac{9}{4}\)

\(\displaystyle \frac{1}{4}x^2 + \frac{3}{2}x + \frac{9}{4}\)

\(\displaystyle \frac{1}{2}x^2 + \frac{3}{2}\)

\(\displaystyle \frac{1}{4}x^2 + 6x + \frac{9}{4}\)

\(\displaystyle \frac{1}{2}x^2 +\frac{1}{2}x+ \frac{3}{2}\)

Correct answer:

\(\displaystyle \frac{1}{4}x^2 + \frac{3}{2}x + \frac{9}{4}\)

Explanation:

The area of a rectangle is:  \(\displaystyle A =LH\)

Substitute the length and height into the formula.

\(\displaystyle A = (x+3)\cdot \frac{1}{4} (x+3)\)

We will move the constant to the front and apply the FOIL method to simplify the binomials.

\(\displaystyle A = \frac{1}{4} (x+3)(x+3) = \frac{1}{4} (x^2+6x+9)\)

Distribute the fraction through all the terms of the trinomial.

The answer is:  \(\displaystyle \frac{1}{4}x^2 + \frac{3}{2}x + \frac{9}{4}\)

Example Question #13 : Geometry

What's the area of triangle with a side of \(\displaystyle \frac{1}{2}x+\frac{1}{4}\) and a height of \(\displaystyle 2x+2\)?

Possible Answers:

\(\displaystyle x^2+3x+1\)

\(\displaystyle \frac{1}{2}x+\frac{1}{4}\)

\(\displaystyle \frac{1}{2}x^2+\frac{3}{4}x+\frac{1}{4}\)

\(\displaystyle x+\frac{1}{2}\)

\(\displaystyle x^2+\frac{3}{2}x+\frac{1}{2}\)

Correct answer:

\(\displaystyle \frac{1}{2}x^2+\frac{3}{4}x+\frac{1}{4}\)

Explanation:

Write the formula for the area of a triangle.

\(\displaystyle A= \frac{BH}{2}\)

Substitute the dimensions.

\(\displaystyle A= \frac{(\frac{1}{2}x+\frac{1}{4})(2x+2)}{2}=\frac{1}{2}(\frac{1}{2}x+\frac{1}{4})(2x+2)\)

\(\displaystyle A= (\frac{1}{2}x+\frac{1}{4})(x+1)\)

Use the FOIL method to expand this.

\(\displaystyle A = (\frac{1}{2}x)(x)+ (\frac{1}{2}x)(1)+ (\frac{1}{4})(x)+ (\frac{1}{4})(1)\)

Simplify the terms.

\(\displaystyle A = \frac{1}{2}x^2+\frac{1}{2}x+\frac{1}{4}x+\frac{1}{4}\)

Combine like-terms.

The answer is:  \(\displaystyle \frac{1}{2}x^2+\frac{3}{4}x+\frac{1}{4}\)

Example Question #14 : Geometry

Determine the side of a square with an area of \(\displaystyle 100x\).

Possible Answers:

\(\displaystyle \sqrt{10x}\)

\(\displaystyle 10x\)

\(\displaystyle 10x^2\)

\(\displaystyle 10000x^2\)

\(\displaystyle 10\sqrt{x}\)

Correct answer:

\(\displaystyle 10\sqrt{x}\)

Explanation:

Write the formula for the area of a square.

\(\displaystyle A = s^2\)

Substitute the area into the equation.

\(\displaystyle 100x= s^2\)

Square root both sides.

\(\displaystyle \sqrt{100x}= \sqrt{s^2}\)

\(\displaystyle s = 10\sqrt{x}\)

The answer is:  \(\displaystyle 10\sqrt{x}\)

Example Question #15 : Geometry

Find the area of a circle with a radius of \(\displaystyle 3x^3\).

Possible Answers:

\(\displaystyle 6\pi x^6\)

\(\displaystyle 6\pi^2 x^6\)

\(\displaystyle 9\pi x^9\)

\(\displaystyle 9\pi^2 x^6\)

\(\displaystyle 9\pi x^6\)

Correct answer:

\(\displaystyle 9\pi x^6\)

Explanation:

The area of a circle is \(\displaystyle A=\pi r^2\).

Substitute the radius and solve for the area.

\(\displaystyle A=\pi (3x^3)^2 = \pi (3x^3)(3x^3) = 9\pi x^6\)

The answer is:  \(\displaystyle 9\pi x^6\)

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