Question 1

Set up an equation and solve the problem. The area of a triangle is 50 square feet. If one side of the triangle and the altitude to that side are of equal length, find the length. __________ feet

Accepted Answer

The answer of Set up an equation and solve the...

Question 2

Set up an equation and solve the problem. The difference of the areas of two squares is 288 square inches. Each side of the larger square is 3 times the length of a side of the smaller square. Find the length of a side of the larger square.

Accepted Answer

Let s be the length of the side of the smaller square. Then the length of the side of the larger square is 3s. The area of the smaller square is s^2 and the area of the larger square is (3s)^2 = 9s^2.

We are told that the difference in area is 288:
9s^2 - s^2 = 288
8s^2 = 288
s^2 = 36
s = 6

So the length of the side of the larger square is 3s = 3(6) = 18 inches. Therefore, the answer is B, 18 in.

Question 3

Set up an equation and solve the problem. 12 times the square of a number equals 588. Find the number.

Accepted Answer

Let's call the number we're trying to find "x". We can set up the equation like this:

12x^2 = 588

To solve for x, we'll first divide both sides by 12:

x^2 = 49

Now we'll take the square root of both sides (remembering that a square root can be positive or negative):

x = ±7

So the possible values for the number we're looking for are -7 or 7, which matches answer choice C.

Question 4

Set up an equation and solve the problem. The area of a triangle is 12.5 square feet. If one side of the triangle and the altitude to that side are of equal length, find the length.

Accepted Answer

Let x be the length of the side and altitude.
Then, the area of the triangle can be expressed as:
(1/2)*x*x = 12.5
x^2 = 25
x = 5

Therefore, the length of the side and altitude is 5 feet.

Question 5

Solve the equation. $2 x ^ { 3 } - 72 x = 0$

Accepted Answer

Factor out $2x$ to get $2x(x^2-36)=0$. Then, $2x=0$ gives $x=0$. Solving $x^2-36=0$ gives $x=\pm6$, so the solutions are $\{-6,0,6\}$. Thus, the answer is choice (C).

Question 6

Set up an equation and solve the problem. The area of a triangle is 40.5 square feet. If one side of the triangle and the altitude to that side are of equal length, find the length.

Accepted Answer

The area of a triangle is given by the formula: Area = 0.5 * base * height. Here, the base and height are equal, so let x be the length of both. The equation becomes 0.5 * x * x = 40.5. Solving for x, we get x^2 = 81, so x = 9 feet.

Question 7

Set up an equation and solve the problem. 144 less than the square of a number equals zero. Find the number.

Accepted Answer

Let x be the number.

Then the equation becomes:

$x^2 - 144 = 0$

Factoring the left side of the equation, we get:

$(x + 12)(x - 12) = 0$

Using the zero product property, we find:

$x + 12 = 0$ or $x - 12 = 0$

So,

$x = -12$ or $x = 12$

Therefore, the solution set is { $-$ 12, 12 }, which corresponds to answer choice D.

Question 8

Set up an equation and solve the problem. The sum of the areas of two squares is 4,477 square inches. Each side of the larger square is 6 times the length of a side of the smaller square. Find the length of a side of the smaller square. __________ inches

Accepted Answer

The answer of Set up an equation and solve the...

Question 9

Set up an equation and solve the problem. The total surface area of a right circular cylinder is 36 $\pi$ square centimeters. If a radius of the base and the altitude of the cylinder are the same length, find the length of a radius.

Accepted Answer

Let r be the radius and h be the altitude of the cylinder.
The formula for the total surface area of a right circular cylinder is: A = 2$\pi$r(r + h)
Since the radius and the altitude are the same length, we have: A = 2$\pi$r(2r) = 4$\pi$r^2
We are given that the total surface area is 36$\pi$, so we can set up an equation:
4$\pi$r^2 = 36$\pi$
Divide both sides by 4$\pi$:
r^2 = 9
Take the square root of both sides:
r = 3 or r = -3
Since radius has to be a positive length, the only valid solution is r = 3. Therefore, the length of the radius is 3 centimeters.

Question 10

Solve the equation. $x ^ { 2 } - 324 = 0$

Accepted Answer

Using the square root property, we get:
$x^2=324$
$x=\pm\sqrt{324}$
$x=\pm18$
Option D is the only one that includes both possible values of x.

Question 11

Set up an equation and solve the problem. Suppose that the width of a certain rectangle is 0.3 of its length and the area of that same rectangle is 24.3 square meters. Find the length of the rectangle.

Accepted Answer

Let the length be L meters. Then the width is 0.3L meters. The area is given by length × width = L × 0.3L = 24.3. Solving 0.3L^2 = 24.3 gives L^2 = 81, so L = 9 meters.

Question 12

Solve the equation. $612 x ^ { 3 } = 17 x$

Accepted Answer

The answer of Solve the equation. \(612 x ^ {...

Question 13

Set up an equation and solve the problem. The sum of the areas of two circles is 160 $\pi$ square meters. Find the length of a radius of the smaller circle if one of them is 3 times as long as the other.

Accepted Answer

Let the radius of the smaller circle be $ r $. Then the radius of the larger circle is $ 3r $. The area of the smaller circle is $ \pi r^2 $ and the area of the larger circle is $ \pi (3r)^2 = 9\pi r^2 $. The sum of the areas is $ \pi r^2 + 9\pi r^2 = 10\pi r^2 = 160\pi $. Solving for $ r $, we get $ 10r^2 = 160 $, so $ r^2 = 16 $, and $ r = 4 $ meters.

Question 14

Set up an equation and solve the problem. Suppose that the width of a certain rectangle is 0.6 of its length and the area of that same rectangle is 0.6 square meters. Find the length of the rectangle. __________ meters

Accepted Answer

The answer of Set up an equation and solve the...

Question 15

Set up an equation and solve the problem. The sum of the areas of two squares is 1,813 square inches. Each side of the larger square is 6 times the length of a side of the smaller square. Find the length of a side of the smaller square.

Accepted Answer

Let x be the side length of the smaller square. Then the side length of the larger square is 6x. The area of the smaller square is x^2, and the area of the larger square is (6x)^2 = 36x^2. The sum of the areas is x^2 + 36x^2 = 37x^2 = 1813. Solving for x, we get x^2 = 49, so x = 7 inches.

Question 16

Set up an equation and solve the problem. The difference of the areas of two squares is 6,912 square inches. Each side of the larger square is 7 times the length of a side of the smaller square. Find the length of a side of the larger square. __________ inches

Accepted Answer

The answer of Set up an equation and solve the...

Question 17

Set up an equation and solve the problem. The total surface area of a right circular cylinder is 100 $\pi$ square centimeters. If a radius of the base and the altitude of the cylinder are the same length, find the length of a radius. __________ centimeters

Accepted Answer

The answer of Set up an equation and solve the...

Question 18

Solve the equation. $n ^ { 3 } = 9 n$

Accepted Answer

First, factor out an $n$ to get $n(n^2 - 9) = 0$, which simplifies to $n(n - 3)(n + 3) = 0$. The solutions are $n = 0$, $n = 3$, and $n = -3$.

Question 19

Solve the equation. $2 x ^ { 2 } = 200$

Accepted Answer

Divide both sides by 2 to get $x^2=100$. Then take the square root of both sides, remembering to include both the positive and negative roots, to get $x=\pm 10$.

Question 20

Set up an equation and solve the problem. The sum of the areas of two circles is 612 $\pi$ square meters. Find the length of a radius of the smaller circle if one of them is 4 times as long as the other. __________ meters

Accepted Answer

The answer of Set up an equation and solve the...

Quiz 7: Factoring, Solving Equations and Problem Solving

Set up an equation and solve the problem. The area of a triangle is 50 square feet. If one side of the triangle and the altitude to that side are of equal length, find the length. __________ feet

Set up an equation and solve the problem. The difference of the areas of two squares is 288 square inches. Each side of the larger square is 3 times the length of a side of the smaller square. Find the length of a side of the larger square.

Set up an equation and solve the problem. 12 times the square of a number equals 588. Find the number.

Set up an equation and solve the problem. The area of a triangle is 12.5 square feet. If one side of the triangle and the altitude to that side are of equal length, find the length.

Solve the equation. 2x3−72x=02 x ^ { 3 } - 72 x = 02x3−72x=0

Set up an equation and solve the problem. The area of a triangle is 40.5 square feet. If one side of the triangle and the altitude to that side are of equal length, find the length.

Set up an equation and solve the problem. 144 less than the square of a number equals zero. Find the number.

Set up an equation and solve the problem. The sum of the areas of two squares is 4,477 square inches. Each side of the larger square is 6 times the length of a side of the smaller square. Find the length of a side of the smaller square. __________ inches

Set up an equation and solve the problem. The total surface area of a right circular cylinder is 36 π\piπ square centimeters. If a radius of the base and the altitude of the cylinder are the same length, find the length of a radius.

Solve the equation. x2−324=0x ^ { 2 } - 324 = 0x2−324=0

Set up an equation and solve the problem. Suppose that the width of a certain rectangle is 0.3 of its length and the area of that same rectangle is 24.3 square meters. Find the length of the rectangle.

Solve the equation. 612x3=17x612 x ^ { 3 } = 17 x612x3=17x

Set up an equation and solve the problem. The sum of the areas of two circles is 160 π\piπ square meters. Find the length of a radius of the smaller circle if one of them is 3 times as long as the other.

Set up an equation and solve the problem. Suppose that the width of a certain rectangle is 0.6 of its length and the area of that same rectangle is 0.6 square meters. Find the length of the rectangle. __________ meters

Set up an equation and solve the problem. The sum of the areas of two squares is 1,813 square inches. Each side of the larger square is 6 times the length of a side of the smaller square. Find the length of a side of the smaller square.

Set up an equation and solve the problem. The difference of the areas of two squares is 6,912 square inches. Each side of the larger square is 7 times the length of a side of the smaller square. Find the length of a side of the larger square. __________ inches

Set up an equation and solve the problem. The total surface area of a right circular cylinder is 100 π\piπ square centimeters. If a radius of the base and the altitude of the cylinder are the same length, find the length of a radius. __________ centimeters

Solve the equation. n3=9nn ^ { 3 } = 9 nn3=9n

Solve the equation. 2x2=2002 x ^ { 2 } = 2002x2=200

Set up an equation and solve the problem. The sum of the areas of two circles is 612 π\piπ square meters. Find the length of a radius of the smaller circle if one of them is 4 times as long as the other. __________ meters

Solve the equation. $2 x ^ { 3 } - 72 x = 0$

Set up an equation and solve the problem. The total surface area of a right circular cylinder is 36 $\pi$ square centimeters. If a radius of the base and the altitude of the cylinder are the same length, find the length of a radius.

Solve the equation. $x ^ { 2 } - 324 = 0$

Solve the equation. $612 x ^ { 3 } = 17 x$

Set up an equation and solve the problem. The sum of the areas of two circles is 160 $\pi$ square meters. Find the length of a radius of the smaller circle if one of them is 3 times as long as the other.

Set up an equation and solve the problem. The total surface area of a right circular cylinder is 100 $\pi$ square centimeters. If a radius of the base and the altitude of the cylinder are the same length, find the length of a radius. __________ centimeters

Solve the equation. $n ^ { 3 } = 9 n$

Solve the equation. $2 x ^ { 2 } = 200$

Set up an equation and solve the problem. The sum of the areas of two circles is 612 $\pi$ square meters. Find the length of a radius of the smaller circle if one of them is 4 times as long as the other. __________ meters