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Sagot :
To determine the amount of paper needed to wrap 6 mini muffins, we first need to find the surface area of one cylindrical mini muffin. Given are the measurements that each muffin has a diameter of 2 inches and a height of [tex]\(1 \frac{1}{2} = 1.5\)[/tex] inches. We will use [tex]\(\pi = \frac{22}{7}\)[/tex].
Step 1: Calculate the radius of the muffin.
Since the diameter of the muffin is 2 inches, the radius [tex]\( r \)[/tex] is half of the diameter:
[tex]\[ r = \frac{2}{2} = 1 \text{ inch} \][/tex]
Step 2: Calculate the lateral surface area.
The lateral surface area of a cylinder is given by the formula:
[tex]\[ \text{Lateral Surface Area} = 2 \pi r h \][/tex]
Substituting the values we have:
[tex]\[ \text{Lateral Surface Area} = 2 \cdot \frac{22}{7} \cdot 1 \cdot 1.5 \][/tex]
[tex]\[ \text{Lateral Surface Area} = 2 \cdot \frac{22}{7} \cdot 1.5 \][/tex]
[tex]\[ \text{Lateral Surface Area} = 2 \cdot \frac{22 \cdot 1.5}{7} \][/tex]
[tex]\[ \text{Lateral Surface Area} = 2 \cdot \frac{33}{7} \][/tex]
[tex]\[ \text{Lateral Surface Area} = \frac{66}{7} \approx 9.43 \text{ square inches} \][/tex]
Step 3: Calculate the area of the top and bottom circles.
The area of one circle is given by the formula:
[tex]\[ \text{Area of one circle} = \pi r^2 \][/tex]
Since there are two circles (top and bottom), we multiply by 2:
[tex]\[ \text{Total Area of Top and Bottom} = 2 \pi r^2 \][/tex]
Substituting the values we have:
[tex]\[ \text{Total Area of Top and Bottom} = 2 \cdot \frac{22}{7} \cdot 1^2 \][/tex]
[tex]\[ \text{Total Area of Top and Bottom} = 2 \cdot \frac{22}{7} \][/tex]
[tex]\[ \text{Total Area of Top and Bottom} = \frac{44}{7} \approx 6.29 \text{ square inches} \][/tex]
Step 4: Calculate the total surface area of one muffin.
The total surface area of one muffin is the sum of the lateral surface area and the total area of the top and bottom:
[tex]\[ \text{Surface Area of One Muffin} = 9.43 + 6.29 \approx 15.72 \text{ square inches} \][/tex]
Step 5: Calculate the total surface area for 6 muffins.
[tex]\[ \text{Total Surface Area for 6 Muffins} = 15.72 \times 6 \approx 94.29 \text{ square inches} \][/tex]
Given the approximate values provided:
1. [tex]\( 94 \frac{2}{7} \text{ in}^2 \)[/tex]
2. [tex]\( 56 \frac{4}{7} \text{ in}^2 \)[/tex]
3. [tex]\( 47 \frac{1}{7} \text{ in}^2 \)[/tex]
4. [tex]\( 15 \frac{5}{7} \text{ in}^2 \)[/tex]
We observe that the closest approximation to our calculated total surface area of [tex]\( 94.29 \text{ square inches} \)[/tex] is [tex]\( 94 \frac{2}{7} \text{ in}^2 \)[/tex].
Therefore, the amount of paper needed to wrap 6 mini muffins is approximately:
[tex]\[ 94 \frac{2}{7} \text{ square inches} \][/tex]
Step 1: Calculate the radius of the muffin.
Since the diameter of the muffin is 2 inches, the radius [tex]\( r \)[/tex] is half of the diameter:
[tex]\[ r = \frac{2}{2} = 1 \text{ inch} \][/tex]
Step 2: Calculate the lateral surface area.
The lateral surface area of a cylinder is given by the formula:
[tex]\[ \text{Lateral Surface Area} = 2 \pi r h \][/tex]
Substituting the values we have:
[tex]\[ \text{Lateral Surface Area} = 2 \cdot \frac{22}{7} \cdot 1 \cdot 1.5 \][/tex]
[tex]\[ \text{Lateral Surface Area} = 2 \cdot \frac{22}{7} \cdot 1.5 \][/tex]
[tex]\[ \text{Lateral Surface Area} = 2 \cdot \frac{22 \cdot 1.5}{7} \][/tex]
[tex]\[ \text{Lateral Surface Area} = 2 \cdot \frac{33}{7} \][/tex]
[tex]\[ \text{Lateral Surface Area} = \frac{66}{7} \approx 9.43 \text{ square inches} \][/tex]
Step 3: Calculate the area of the top and bottom circles.
The area of one circle is given by the formula:
[tex]\[ \text{Area of one circle} = \pi r^2 \][/tex]
Since there are two circles (top and bottom), we multiply by 2:
[tex]\[ \text{Total Area of Top and Bottom} = 2 \pi r^2 \][/tex]
Substituting the values we have:
[tex]\[ \text{Total Area of Top and Bottom} = 2 \cdot \frac{22}{7} \cdot 1^2 \][/tex]
[tex]\[ \text{Total Area of Top and Bottom} = 2 \cdot \frac{22}{7} \][/tex]
[tex]\[ \text{Total Area of Top and Bottom} = \frac{44}{7} \approx 6.29 \text{ square inches} \][/tex]
Step 4: Calculate the total surface area of one muffin.
The total surface area of one muffin is the sum of the lateral surface area and the total area of the top and bottom:
[tex]\[ \text{Surface Area of One Muffin} = 9.43 + 6.29 \approx 15.72 \text{ square inches} \][/tex]
Step 5: Calculate the total surface area for 6 muffins.
[tex]\[ \text{Total Surface Area for 6 Muffins} = 15.72 \times 6 \approx 94.29 \text{ square inches} \][/tex]
Given the approximate values provided:
1. [tex]\( 94 \frac{2}{7} \text{ in}^2 \)[/tex]
2. [tex]\( 56 \frac{4}{7} \text{ in}^2 \)[/tex]
3. [tex]\( 47 \frac{1}{7} \text{ in}^2 \)[/tex]
4. [tex]\( 15 \frac{5}{7} \text{ in}^2 \)[/tex]
We observe that the closest approximation to our calculated total surface area of [tex]\( 94.29 \text{ square inches} \)[/tex] is [tex]\( 94 \frac{2}{7} \text{ in}^2 \)[/tex].
Therefore, the amount of paper needed to wrap 6 mini muffins is approximately:
[tex]\[ 94 \frac{2}{7} \text{ square inches} \][/tex]
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