Question
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?
meters by
meters
meters by
meters
Answers
Answer
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
feet by
feet
Answer
A square notepad has a perimeter of 132 centimeters. How long is each side of the notepad?
​
centimeters
​
centimeters
Answer
A square painting has sides that are 7 feet long. What is the painting's area?
​
square feet
​
square feet
Answer
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
A rectangular ceiling tile is 40 inches long and 13 inches wide. What is its perimeter?
​
inches
​
inches
Answered by
GPT-4o mini
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 \(\mathbf{10 \text{ meters by } 8 \text{ meters}}\).
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 \(\mathbf{10 \text{ meters by } 8 \text{ meters}}\).
Answered by
GPT-4o mini
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 \(\mathbf{5 \text{ feet by } 3 \text{ feet}}\).
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 \(\mathbf{5 \text{ feet by } 3 \text{ feet}}\).
Answered by
GPT-4o mini
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:
\[
\mathbf{33 \text{ centimeters}}
\]
\[
\text{Perimeter} = 4s
\]
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:
\[
\mathbf{33 \text{ centimeters}}
\]
Answered by
GPT-4o mini
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:
\[
\mathbf{49 \text{ square feet}}
\]
\[
A = s^2
\]
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:
\[
\mathbf{49 \text{ square feet}}
\]
Answered by
GPT-4o mini
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:
\[
\mathbf{\$228.00}
\]
\[
P = 4s
\]
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:
\[
\mathbf{\$228.00}
\]
Answered by
GPT-4o mini
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:
\[
\mathbf{106 \text{ inches}}
\]
\[
P = 2l + 2w
\]
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:
\[
\mathbf{106 \text{ inches}}
\]
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