The area of a rectangular shop in the mall is 80 square meters. The perimeter is 36 meters. What are the dimensions of the shop?

Question

meters by
meters

Answer 1

To find the dimensions of the rectangular shop, we can use the formulas for area and perimeter.

Let the length of the shop be \( l \) meters and the width be \( w \) meters.

We have two equations based on the given information:

1. Area: \[ l \times w = 80 \]

2. Perimeter: \[ 2l + 2w = 36 \] This simplifies to: \[ l + w = 18 \]

Now we have a system of two equations:

1. \( l \times w = 80 \)
2. \( l + w = 18 \)

We can express \( w \) in terms of \( l \) using the second equation: \[ w = 18 - l \]

Now, substitute \( w \) into the first equation: \[ l \times (18 - l) = 80 \] Expanding this gives: \[ 18l - l^2 = 80 \] Rearranging it results in a standard quadratic equation: \[ l^2 - 18l + 80 = 0 \]

We can solve this quadratic equation using the quadratic formula: \[ l = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 1 \), \( b = -18 \), and \( c = 80 \).

Calculating the discriminant: \[ b^2 - 4ac = (-18)^2 - 4(1)(80) = 324 - 320 = 4 \]

Now, substituting into the quadratic formula: \[ l = \frac{18 \pm \sqrt{4}}{2} \] \[ l = \frac{18 \pm 2}{2} \]

This gives us two possible solutions: \[ l = \frac{20}{2} = 10 \quad \text{or} \quad l = \frac{16}{2} = 8 \]

If \( l = 10 \), then: \[ w = 18 - l = 18 - 10 = 8 \]

If \( l = 8 \), then: \[ w = 18 - l = 18 - 8 = 10 \]

Thus, the dimensions of the shop are: \[ 10 \text{ meters by } 8 \text{ meters} \]

So, the dimensions are \(\mathbf{10 \text{ meters by } 8 \text{ meters}}\).

Answer 2

A rectangular painting has a perimeter of 16 feet and an area of 15 square feet. What are the dimensions of the painting?

feet by
feet

Answer 3

To find the dimensions of the rectangular painting, we can use the formulas for perimeter and area.

Let the length of the painting be \( l \) feet and the width be \( w \) feet.

We have two equations based on the given information:

1. Area: \[ l \times w = 15 \]

2. Perimeter: \[ 2l + 2w = 16 \] This simplifies to: \[ l + w = 8 \]

Now we have a system of two equations:

1. \( l \times w = 15 \)
2. \( l + w = 8 \)

We can express \( w \) in terms of \( l \) using the second equation: \[ w = 8 - l \]

Now, substitute \( w \) into the first equation: \[ l \times (8 - l) = 15 \] Expanding this gives: \[ 8l - l^2 = 15 \] Rearranging it results in a standard quadratic equation: \[ l^2 - 8l + 15 = 0 \]

We can solve this quadratic equation using the quadratic formula: \[ l = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 1 \), \( b = -8 \), and \( c = 15 \).

Calculating the discriminant: \[ b^2 - 4ac = (-8)^2 - 4(1)(15) = 64 - 60 = 4 \]

Now substituting into the quadratic formula: \[ l = \frac{8 \pm \sqrt{4}}{2} \] \[ l = \frac{8 \pm 2}{2} \]

This gives us two possible solutions: \[ l = \frac{10}{2} = 5 \quad \text{or} \quad l = \frac{6}{2} = 3 \]

If \( l = 5 \), then: \[ w = 8 - l = 8 - 5 = 3 \]

If \( l = 3 \), then: \[ w = 8 - l = 8 - 3 = 5 \]

Thus, the dimensions of the painting are: \[ 5 \text{ feet by } 3 \text{ feet} \]

So, the dimensions are \(\mathbf{5 \text{ feet by } 3 \text{ feet}}\).

Answer 4

A square notepad has a perimeter of 132 centimeters. How long is each side of the notepad?

​
centimeters

Answer 5

To find the length of each side of a square notepad given its perimeter, we can use the formula for the perimeter of a square, which is:

\[ \text{Perimeter} = 4s \]

where \( s \) is the length of one side.

Given that the perimeter is 132 centimeters, we can set up the equation:

\[ 4s = 132 \]

To find \( s \), divide both sides by 4:

\[ s = \frac{132}{4} = 33 \]

Thus, the length of each side of the notepad is:

\[ \mathbf{33 \text{ centimeters}} \]

Answer 6

A square painting has sides that are 7 feet long. What is the painting's area?

​
square feet

Answer 7

The area \( A \) of a square can be calculated using the formula:

\[ A = s^2 \]

where \( s \) is the length of one side of the square.

For a square painting with sides that are 7 feet long:

\[ A = 7^2 = 49 \]

Thus, the area of the painting is:

\[ \mathbf{49 \text{ square feet}} \]

Answer 8

A square window in Kylie's room has sides that are 19 inches long. Kylie wants to buy a new window frame that costs $3.00 per inch. How much will the window frame for Kylie's window cost?

$

Answer 9

To find the cost of the window frame, we first need to determine the perimeter of the square window. The formula for the perimeter \( P \) of a square is:

\[ P = 4s \]

where \( s \) is the length of one side of the square.

In this case, the length of each side is 19 inches:

\[ P = 4 \times 19 = 76 \text{ inches} \]

Now, to find the cost of the window frame, we multiply the perimeter by the cost per inch. Given that the cost is $3.00 per inch:

\[ \text{Cost} = P \times \text{Cost per inch} = 76 \times 3.00 \]

Calculating this gives:

\[ \text{Cost} = 76 \times 3.00 = 228.00 \]

Thus, the cost of the window frame for Kylie's window will be:

\[ \mathbf{$228.00} \]

Answer 10

A rectangular ceiling tile is 40 inches long and 13 inches wide. What is its perimeter?

​
inches

Answer 11

To find the perimeter \( P \) of a rectangular ceiling tile, we can use the formula:

\[ P = 2l + 2w \]

where \( l \) is the length and \( w \) is the width.

Given that the length \( l \) is 40 inches and the width \( w \) is 13 inches, we can substitute these values into the formula:

\[ P = 2(40) + 2(13) \]

Calculating this:

\[ P = 80 + 26 = 106 \]

Thus, the perimeter of the ceiling tile is:

\[ \mathbf{106 \text{ inches}} \]