Basic Single-Variable Algebra

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Math › Basic Single-Variable Algebra

Questions 1 - 10

varies inversely as the square root of . If , then . Find if (nearest tenth, if applicable).

Explanation

The variation equation is $A = \frac{K}{\sqrt{x}}$ for some constant of variation .

Substitute the numbers from the first scenario to find :

$45= \frac{K}{\sqrt{15}}$

$K = 45 \sqrt{15}$

The equation is now $A = \frac{45 \sqrt{15}}{\sqrt{x}}$ .

If , then

$A = \frac{45 \sqrt{15}}{\sqrt{30}} \approx 31.8$

Set up the equation: Eight less than four times a number is twenty.

Explanation

Split the sentence into parts.

Four times a number:

Eight less than four times a number:

Is twenty:

Set the terms equal.

The answer is:

Set up the equation: Nine more than three times the cube of a number is four.

Explanation

Split the sentence into parts.

The cube of a number:

Three times the cube of a number:

Nine more than three times the cube of a number:

Is four:

Combine the parts to form the equation.

The answer is:

Solve the equation for .

No solution

Explanation

Since we have variables on both sides, we first subtract from both sides.

This leaves us with .

Subtracting 2 from both sides, we get .

x varies inversely with y. When x=10, y=6. When x=3, what is y?

Explanation

Inverse variation takes the form:

$y=\frac{k}{x}$

Plugging in:

$6=\frac{k}{10}$

Then solve when x=3:

Solve the equation:

$\frac{2}{5}$

$\frac{2}{13}$

$-\frac{2}{13}$

$\frac{20}{13}$

Explanation

Add on both sides.

Combine like-terms on both sides.

Subtract nine from both sides.

Divide by five on both sides.

$\frac{5x}{5}=\frac{2}{5}$

The answer is: $\frac{2}{5}$

$\small x$ varies directly with $\small A$ and inversely with the square root of $\small B$ . Find values for $\small A$ and $\small B$ that will give $\small x = 3$ , for a constant of variation $\small k = 2$ .

All of these answers are correct

$\small A = 3$ and $\small B = 4$

$\small A = 7.5$ and $\small B = 25$

$\small A = 9$ and $\small B = 36$

Explanation

From the first sentence, we can write the equation of variation as:

$\small x = k \cdot \frac{A}{\sqrt{B}}$

We can then check each of the possible answer choices by substituting the values into the variation equation with the values given for $\small x$ and $\small k$ .

$\small 3 = 2 \cdot \frac{3}{\sqrt{4}} = 2 \cdot \frac{3}{2} = 3$

Therefore the equation is true if $\small A = 3$ and $\small B = 4$

$\small 3 = 2 \cdot \frac{7.5}{\sqrt{25}} = 2 \cdot \frac{7.5}{5} = 2 \cdot 1.5 = 3$

Therefore the equation is true if $\small A = 7.5$ and $\small B = 25$

$\small 3 = 2 \cdot \frac{9}{\sqrt{36}} = 2 \cdot \frac{9}{6} = 2 \cdot 1.5 = 3$

Therefore the equation is true if $\small A = 9$ and $\small B = 36$

The correct answer choice is then "All of these answers are correct"

varies inversely as the square root of . If , then . Find if (nearest tenth, if applicable).

Explanation

The variation equation is $A = \frac{K}{\sqrt{x}}$ for some constant of variation .

Substitute the numbers from the first scenario to find :

$45= \frac{K}{\sqrt{15}}$

$K = 45 \sqrt{15}$

The equation is now $A = \frac{45 \sqrt{15}}{\sqrt{x}}$ .

If , then

$A = \frac{45 \sqrt{15}}{\sqrt{30}} \approx 31.8$

A circular tower stands surrounded by a circular moat. A bridge provides a passage over the moat to the tower. The distance from the outer edge of the moat to the center of the tower is meters. The area of the floor of tower is $4900\pi$ $m^{2}$ . How long is the bridge over the moat?

Explanation

The distance from the outer edge of the moat to the center of the castle is the radius (100 m) of the larger circle formed by the outer edge of the circular moat.

The radius of the tower's floor (found using the area of the floor), needs to be subracted from 100 m to find the distance fo the bridge.

$\sqrt{4900}=70$