A poster has an area of 10 square feet. Its perimeter is 14 feet. What are the dimensions of the poster?%0D%0A%0D%0A feet by feet%0D%0ASubmit

Answer 1

Let the dimensions of the poster be \( l \) (length) and \( w \) (width). We have the following equations based on the problem statement:

1. Area: \( l \times w = 10 \) (1)
2. Perimeter: \( 2l + 2w = 14 \) (2)

From equation (2), we can simplify it:

\[ l + w = 7 \quad \text{(3)} \]

Now we have a system of equations:

1. \( l \times w = 10 \)
2. \( l + w = 7 \)

We can solve (3) for \( w \):

\[ w = 7 - l \]

Now substitute this expression for \( w \) into (1):

\[ l \times (7 - l) = 10 \]

Expanding this gives:

\[ 7l - l^2 = 10 \]

Rearranging leads to a quadratic equation:

\[ l^2 - 7l + 10 = 0 \]

We can factor this as:

\[ (l - 5)(l - 2) = 0 \]

This gives us two possible values for \( l \):

\[ l = 5 \quad \text{or} \quad l = 2 \]

Using equation (3) to find the corresponding \( w \):

• If \( l = 5 \): \[ w = 7 - 5 = 2 \]

• If \( l = 2 \): \[ w = 7 - 2 = 5 \]

Thus, the dimensions of the poster are:

\[ 5 \text{ feet} \text{ by } 2 \text{ feet} \]

So the final result is:

5 feet by 2 feet.

Answer 2

A rectangular portrait is 2 meters wide and 1 meter high. It costs $4.00 per meter to put a gold frame around the portrait. How much will the frame cost?

Answer 3

To find the cost of the gold frame around the rectangular portrait, we first need to calculate the perimeter of the portrait, since the cost of the frame is given per meter of perimeter.

The formula for the perimeter \( P \) of a rectangle is:

\[ P = 2 \times ( \text{width} + \text{height} ) \]

For the given portrait:

• Width = 2 meters
• Height = 1 meter

Now we can substitute the values into the formula:

\[ P = 2 \times (2 , \text{m} + 1 , \text{m}) = 2 \times 3 , \text{m} = 6 , \text{m} \]

Next, we calculate the cost of the frame. The cost is $4.00 per meter, so the total cost \( C \) can be calculated as follows:

\[ C = 6 , \text{m} \times 4 , \text{USD/m} = 24 , \text{USD} \]

Therefore, the cost of the frame is $24.00.

Answer 4

The perimeter of a bookmark is 20 centimeters. The area is 16 square centimeters. What are the dimensions of the bookmark?

Answer 5

Let the dimensions of the bookmark be \( l \) (length) and \( w \) (width). We have the following equations based on the problem statement:

1. Perimeter: \( 2l + 2w = 20 \) (1)
2. Area: \( l \times w = 16 \) (2)

We can simplify equation (1):

\[ l + w = 10 \quad \text{(3)} \]

Now we have a system of equations to work with:

1. \( l + w = 10 \)
2. \( l \times w = 16 \)

From equation (3), we can express \( w \) in terms of \( l \):

\[ w = 10 - l \]

Now substitute this expression for \( w \) into equation (2):

\[ l \times (10 - l) = 16 \]

Expanding this gives:

\[ 10l - l^2 = 16 \]

Rearranging leads to a quadratic equation:

\[ l^2 - 10l + 16 = 0 \]

We can use the quadratic formula, \( l = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = -10 \), and \( c = 16 \):

\[ b^2 - 4ac = (-10)^2 - 4(1)(16) = 100 - 64 = 36 \]

Now applying the quadratic formula:

\[ l = \frac{10 \pm \sqrt{36}}{2(1)} = \frac{10 \pm 6}{2} \]

This gives us two solutions for \( l \):

1. \( l = \frac{16}{2} = 8 \)
2. \( l = \frac{4}{2} = 2 \)

Using equation (3) to find the corresponding \( w \):

• If \( l = 8 \): \[ w = 10 - 8 = 2 \]

• If \( l = 2 \): \[ w = 10 - 2 = 8 \]

Thus, the dimensions of the bookmark are:

\[ 8 , \text{cm} \text{ by } 2 , \text{cm} \]

So the final result is:

8 cm by 2 cm.

Answer 6

Brandon's pool table is 3 feet wide and 7 feet long. Brandon wants to replace the felt on the pool table. The felt costs $2.00 per square foot. How much would it cost in total to replace the felt on the pool table?

Answer 7

To calculate the cost of replacing the felt on Brandon's pool table, we first need to determine the area of the pool table. The area \( A \) of a rectangle can be calculated using the formula:

\[ A = \text{length} \times \text{width} \]

In this case, the dimensions of the pool table are:

• Length = 7 feet
• Width = 3 feet

Substituting these values into the area formula:

\[ A = 7 , \text{ft} \times 3 , \text{ft} = 21 , \text{square feet} \]

Next, we need to calculate the cost to replace the felt. The cost per square foot is $2.00, so the total cost \( C \) can be calculated as follows:

\[ C = \text{Area} \times \text{Cost per square foot} = 21 , \text{square feet} \times 2 , \text{USD/square foot} \]

Calculating this gives:

\[ C = 21 \times 2 = 42 , \text{USD} \]

Therefore, the total cost to replace the felt on the pool table is $42.00.

Answer 8

Each side of a square courtyard is 21 meters long. It costs $13.78 per meter to replace the brick wall around the courtyard. How much would it cost to replace the brick wall?

Answer 9

To calculate the cost of replacing the brick wall around a square courtyard, we first need to determine the perimeter of the courtyard. The perimeter \( P \) of a square can be calculated using the formula:

\[ P = 4 \times \text{side length} \]

Given that each side of the square courtyard is 21 meters long, we can substitute this value into the formula:

\[ P = 4 \times 21 , \text{m} = 84 , \text{meters} \]

Next, we need to calculate the cost to replace the brick wall. The cost per meter is $13.78, so the total cost \( C \) can be calculated as follows:

\[ C = \text{Perimeter} \times \text{Cost per meter} = 84 , \text{meters} \times 13.78 , \text{USD/meter} \]

Calculating this gives:

\[ C = 84 \times 13.78 = 1155.12 , \text{USD} \]

Therefore, the total cost to replace the brick wall around the courtyard is $1155.12.