### All ACT Math Resources

## Example Questions

### Example Question #1 : How To Find The Perimeter Of A Rectangle

What is the perimeter of the below rectangle in simplest radical form?

**Possible Answers:**

4√3 + 2√27

7√27

10√3

5√3

**Correct answer:**

10√3

The perimeter of a figure is the sum of the lengths of all of its sides. The perimeter of this figure is √27 + 2√3 + √27 + 2√3. But, √27 = √9√3 = 3√3 . Now all of the sides have the same number underneath of the radical symbol (i.e. the same radicand) and so the coefficients of each radical can be added together. The result is that the perimeter is equal to 10√3.

### Example Question #5 : How To Find The Perimeter Of A Rectangle

A rectangle has an area of 56 square feet, and a width of 4 feet. What is the perimeter, in feet, of the rectangle?

**Possible Answers:**

**Correct answer:**36

Divide the area of the rectangle by the width in order to find the length of 14 feet. The perimeter is the sum of the side lengths, which in this case is 14 feet + 4 feet +14 feet + 4 feet, or 36 feet.

### Example Question #1 : How To Find The Perimeter Of A Rectangle

The length of a rectangle is 3 more inches than its width. The area of the rectangle is 40 in^{2}. What is the perimeter of the rectangle?

**Possible Answers:**

None of the answers are correct

34 in.

40 in.

26 in.

18 in.

**Correct answer:**

26 in.

Area of rectangle: A = lw

Perimeter of rectangle: P = 2l + 2w

w = width and l = w + 3

So A = w(w + 3) = 40 therefore w^{2} + 3w – 40 = 0

Factor the quadratic equation and set each factor to 0 and solve.

w^{2} + 3w – 40 = (w – 5)(w + 8) = 0 so w = 5 or w = -8.

The only answer that makes sense is 5. You cannot have a negative value for a length.

Therefore, w = 5 and l = 8, so P = 2l + 2w = 2(8) + 2(5) = 26 in.

### Example Question #1 : Rectangles

Kayla took 25 minutes to walk around a rectangular city block. If the block's width is 1/4 the size of the length, how long would it take to walk along one length?

**Possible Answers:**

10 minutes

8 minutes

7 minutes

2.5 minutes

**Correct answer:**

10 minutes

Leaving the width to be *x*, the length is 4*x*. The total perimeter is 4*x* + 4*x* + *x* + *x *= 10x.

We divide 25 by 10 to get 2.5, the time required to walk the width. Therefore the time required to walk the length is (4)(2.5) = 10.

### Example Question #1 : How To Find The Perimeter Of A Rectangle

The area of a rectangle is , and the width of this rectangle is two times its height. What is the perimeter of the rectangle?

**Possible Answers:**

**Correct answer:**

The area of a rectangle is the width times the height, and we are told that in this rectangle the width is two times the height.

Therefore, .

Plug in the value of the area:

Solve for to find a width of 4 inches. Using our formula above, the height must be 8 inches. We then add all the sides of the rectangle together to find the perimeter:

### Example Question #2 : How To Find The Perimeter Of A Rectangle

Robert is designing a rectangular garden. He wants the area of the garden to be 9 square meters. If the length of the lot is going to be three meters less than twice the width, what will the perimeter of the lot be in meters?

**Possible Answers:**

6

1.5

3

10

12

**Correct answer:**

12

Let l be the length of the garden and w be the width.

By the specifications of the problem, l = 2w-3.

Plug this in for length in the area formula:

A = l x w = (2w - 3) x w = 9

Solve for the width:

2w²- 3w - 9 =0

(2w + 3)(w - 3) = 0

w is either 3 or -3/2, but we can't have a negative width, so w = 3.

If w = 3, then length = 2(3) - 3 = 3.

Now plug the width and length into the formula for perimeter:

P = 2 l + 2w = 2(3) + 2(3) = 12

### Example Question #1 : How To Find The Perimeter Of A Rectangle

What is the perimeter of a rectangle with one side of and an area of ?

**Possible Answers:**

**Correct answer:**

Based on the information provided, you know that the rectangle's area could be represented as:

, where is the unknown side of the rectangle. Solving for , you get .

Now, remember that your perimeter is just the sum of all the sides. This is:

For our data, this is:

### Example Question #2 : How To Find The Perimeter Of A Rectangle

The area of a yard is . If one side is in length, how many feet of fence would be needed to surround the yard?

**Possible Answers:**

**Correct answer:**

Begin by computing the length of the other side of the yard. Based on the data that you have, you know that:

, where is the other side in yards. Solving for is:

Now, this means that our yard's dimensions in feet are:

The perimeter will be . For our data, this is:

### Example Question #1 : How To Find The Perimeter Of A Rectangle

A person spends on fence for his yard, spending . What is the larger dimension of his yard if the total area of the yard is ?

**Possible Answers:**

**Correct answer:**

Based on this data, you know that the person purchased a total of or of fence. Based on all of the data you have, you can write the two equations:

Solve the first equation for :

Now substitute this into the first equation:

Since this is a quadratic equation, isolate everything to one side:

This is a bit tricky to factor, but it comes out as:

This means that the larger dimension is .

### Example Question #2 : How To Find The Perimeter Of A Rectangle

A rectangle has an area of and one side has a length of , what is the perimeter of the rectangle?

**Possible Answers:**

**Correct answer:**

To find the length and width of a rectangle given the area and one side, simply divide the area by the given side length:

Next, to find the perimeter use the formula: