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SAT Math : SAT Mathematics

Study concepts, example questions & explanations for SAT Math

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Example Questions

Example Question #1 : How To Use The Inverse Variation Formula

The radius of the base of a cylinder is ; the height of the same cylinder is ; the cylinder has volume 1,000.

$t = \sqrt{r}$

$v = \sqrt[4]{h}$

Which of the following is a true statement?

Assume all quantities are positive.

Possible Answers:

varies directly as the square of .

varies inversely as the square root of .

varies inversely as .

varies dorectly as the square root of .

varies inversely as the square of .

Correct answer:

varies inversely as .

Explanation:

The volume of a cylinder can be calculated from its height and the radius of its base using the formula:

$\pi r^{2} h = V$

$t = \sqrt{r}$ , so $t ^{2}= r$ ;

$v = \sqrt[4]{h}$ , so $h = v^{4}$ .

The volume is 1,000, and by substitution, using the other equations:

$\pi (t^{2})^{2} v^{4} = 1,000$

$\pi t^{4} v^{4} = 1,000$

$\frac{\pi t^{4} v^{4}}{\pi} = \frac{1,000}{\pi}$

$t^{4} v^{4} = \frac{1,000}{\pi}$

$\sqrt[4]{t^{4} v^{4} }= \sqrt[4]{\frac{1,000}{\pi}}$

$tv= \sqrt[4]{\frac{1,000}{\pi}}$

If we take $K= \sqrt[4]{\frac{1,000}{\pi}}$ as the constant of variation, we get

tv=K ,

meaning that varies inversely as .

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Example Question #3 : How To Use The Inverse Variation Formula

varies directly as the square of and the cube root of , and inversely as the fourth root of . Which of the following is a true statement?

Possible Answers:

varies directly as the square root of and the eighth root of , and inversely as the sixth power of .

varies directly as the square root of and the eighth root of , and inversely as the sixth root of .

varies directly as the square root of and the eighth power of , and inversely as the sixth root of .

varies directly as the square root of , and inversely as the eighth root of and the sixth root of .

varies directly as the square root of , and inversely as the eighth power of and the sixth power of .

Correct answer:

varies directly as the square root of and the eighth root of , and inversely as the sixth root of .

Explanation:

varies directly as the square of and the cube root of , and inversely as the fourth root of , so, for some constant of variation ,

$\frac{s \sqrt[4]{w}}{y^{2}\sqrt[3]{t}} = K$

We take the reciprocal of both sides, then extract the square root:

$\frac{y^{2}\sqrt[3]{t}}{s \sqrt[4]{w}} =\frac{1} { K}$

$\frac{y^{2}t^{\frac{1}{3}}}{s w ^{\frac{1}{4}}} =\frac{1} { K}$

$\left ( \frac{y^{2}t^{\frac{1}{3}}}{s w ^{\frac{1}{4}}} \right )^{\frac{1}{2}} =\left (\frac{1} { K} \right )^{\frac{1}{2}}$

$\frac{y t^{\frac{1}{6}}}{s ^{\frac{1}{2}} w ^{\frac{1}{8}}}=\left (\frac{1} { K} \right )^{\frac{1}{2}}$

$\frac{y\sqrt[6]{t}}{\sqrt{s} \cdot \sqrt[8]{w}}=\left (\frac{1} { K} \right )^{\frac{1}{2}}$

Taking $\left (\frac{1} { K} \right )^{\frac{1}{2}}$ as the constant of variation, we see that varies directly as the square root of and the eighth root of , and inversely as the sixth root of .

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Example Question #2 : How To Use The Inverse Variation Formula

If varies inversely as , and y=14 when x=3 , find when x=28 .

Possible Answers:

$\frac{4}{7}$

$1 \frac{5}{7}$

$1\frac{1}{2}$

$37 \frac{1}{3}$

Correct answer:

$1\frac{1}{2}$

Explanation:

The formula for inverse variation is as follows: $y=\frac{k}{x}$

Use the x and y values from the first part of the sentence to find k.

$14=\frac{k}{3}$

k=42

Then use that k value and the given x value to find y.

$y=\frac{42}{28}$

$y=1\frac{1}{2}$

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Example Question #1 : How To Find The Degree Of A Polynomial

Find the degree of the polynomial:

$x^{3}-3x^{5}+7$

Possible Answers:

Correct answer:

Explanation:

To find the degree of a polynomial we must find the largest exponent in the function.

The degree of the polynomial $x^{3}-3x^{5}+7$ is 5, as the largest exponent of is 5 in the second term.

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Example Question #1 : Polynomial Operations

What is the degree of the polynomial $5x^{2}y^{2}+4y^{3}+3x+10$ ?

Possible Answers:

Correct answer:

Explanation:

When a polynomial has more than one variable, we need to find the degree by adding the exponents of each variable in each term.

$5x^{2}y^{2}$ has a degree of 4 (since both exponents add up to 4), so the polynomial has a degree of 4 as this term has the highest degree.

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Example Question #3 : How To Find The Degree Of A Polynomial

Find the degree of the following polynomial:

$7xy^{2}z^{3}+8y^{5}-3x^{4}+4$

Possible Answers:

Correct answer:

Explanation:

When a polynomial has more than one variable, we need to find the degree by adding the exponents of each variable in each term.

Even though $8y^{5}$ has a degree of 5, it is not the highest degree in the polynomial -

$7xy^{2}z^{3}$ has a degree of 6 (with exponents 1, 2, and 3). Therefore, the degree of the polynomial is 6.

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Example Question #1 : How To Find The Degree Of A Polynomial

Solve each problem and decide which is the best of the choices given.

What is the degree of the following polynomial?

3x^4-9x^2+12x^7-9x

Possible Answers:

Correct answer:

Explanation:

The degree is defined as the largest exponent in the polynomial. In this case, it is .

$3x^4-9x^2+12x^{\color{Red} 7}-9x$

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Example Question #1 : Polynomials

What is the degree of this polynomial?

x^2(x^3)^2

Possible Answers:

Degree 10

Degree 6

Degree 12

Degree 8

Degree 7

Correct answer:

Degree 8

Explanation:

x^2(x^3)^2

When an exponent with a power is raised to another power, the value of the power are multiplied.

$x^2(x^{2*3})=x^2(x^6)$

When multiplying exponents you add the powers together

$x^{6+2}=x^8$

The degree of a polynomial is the determined by the highest power. In this problem the highest power is 8.

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Example Question #1 : Polynomial Operations

Find the degree of the following polynomial: 12-x^5+18x^3-100x^2+3x^6

Possible Answers:

Correct answer:

Explanation:

The degree of a polynomial is the largest exponent on one of its variables (for a single variable), or the largest sum of exponents on variables in a single term (for multiple variables).

Here, the term with the largest exponent is 3x^6 , so the degree of the whole polynomial is 6.

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Example Question #1 : Polynomial Operations

If 3 less than 15 is equal to 2x, then 24/x must be greater than

Possible Answers:

Correct answer:

Explanation:

Set up an equation for the sentence: 15 – 3 = 2x and solve for x. X equals 6. If you plug in 6 for x in the expression 24/x, you get 24/6 = 4. 4 is only choice greater than a.

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SAT Math : SAT Mathematics

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #1 : How To Use The Inverse Variation Formula

Example Question #3 : How To Use The Inverse Variation Formula

Example Question #2 : How To Use The Inverse Variation Formula

Example Question #1 : How To Find The Degree Of A Polynomial

Example Question #1 : Polynomial Operations

Example Question #3 : How To Find The Degree Of A Polynomial

Example Question #1 : How To Find The Degree Of A Polynomial

Example Question #1 : Polynomials

Example Question #1 : Polynomial Operations

Example Question #1 : Polynomial Operations

All SAT Math Resources

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