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Sagot :
Sure, let's break down each part of the question and work through the steps.
### Given:
- Diameter of the cone = 3.5 inches
- Height of the cone = 8 inches
### Solution:
#### Part a: Surface area of the cone (excluding the base)
To find the surface area of a cone (excluding the base), we need to use the formula for the lateral surface area:
[tex]\[ \text{Lateral Surface Area} = \pi \times r \times l \][/tex]
where [tex]\( r \)[/tex] is the radius of the cone and [tex]\( l \)[/tex] is the slant height.
1. Calculate the radius:
[tex]\[ r = \frac{\text{diameter}}{2} = \frac{3.5}{2} = 1.75 \text{ inches} \][/tex]
2. Calculate the slant height [tex]\( l \)[/tex]:
[tex]\[ l = \sqrt{r^2 + h^2} = \sqrt{1.75^2 + 8^2} = \sqrt{3.0625 + 64} = \sqrt{67.0625} = 8.19 \text{ inches} \][/tex]
3. Calculate the lateral surface area:
[tex]\[ \text{Lateral Surface Area} = \pi \times 1.75 \times 8.19 = 45.02 \text{ square inches} \][/tex]
4. Round to the nearest square inch:
[tex]\[ \text{Rounded Surface Area} = 45 \text{ square inches} \][/tex]
So, the waffle cone takes approximately [tex]\( 45 \)[/tex] square inches of waffle material to make.
#### Part b: Volume of the cone
To find the volume of a cone, we use the formula:
[tex]\[ \text{Volume} = \frac{1}{3} \pi r^2 h \][/tex]
1. Calculate the volume:
[tex]\[ \text{Volume} = \frac{1}{3} \pi (1.75)^2 \times 8 = \frac{1}{3} \pi \times 3.0625 \times 8 = \frac{1}{3} \pi \times 24.5 = 25.66 \text{ cubic inches} \][/tex]
2. Round to the nearest cubic inch:
[tex]\[ \text{Rounded Volume} = 26 \text{ cubic inches} \][/tex]
So, the cone can hold approximately [tex]\( 26 \)[/tex] cubic inches of ice cream.
#### Part c: Number of spherical scoops
Next, we need to find out how many perfectly spherical scoops of ice cream (with diameter 3.5 inches) can fit into the cone.
1. Calculate the volume of a spherical scoop:
[tex]\[ \text{Volume} = \frac{4}{3} \pi r^3 \][/tex]
Since the diameter of the scoop is 3.5 inches, the radius is:
[tex]\[ r = \frac{3.5}{2} = 1.75 \text{ inches} \][/tex]
[tex]\[ \text{Volume} = \frac{4}{3} \pi (1.75)^3 = \frac{4}{3} \pi \times 5.359375 = 22.45 \text{ cubic inches} \][/tex]
2. Calculate the number of scoops:
[tex]\[ \text{Number of scoops} = \frac{\text{Volume of cone}}{\text{Volume of one scoop}} = \frac{25.66}{22.45} = 1.14 \][/tex]
3. Round to the nearest whole number:
[tex]\[ \text{Rounded Number of Scoops} = 1 \][/tex]
So, approximately 1 spherical scoop of ice cream could melt into the cone without overflowing.
### Summary of Answers:
a. [tex]\( 45 \)[/tex] square inches
b. [tex]\( 26 \)[/tex] cubic inches
c. [tex]\( 1 \)[/tex] spherical scoop
### Given:
- Diameter of the cone = 3.5 inches
- Height of the cone = 8 inches
### Solution:
#### Part a: Surface area of the cone (excluding the base)
To find the surface area of a cone (excluding the base), we need to use the formula for the lateral surface area:
[tex]\[ \text{Lateral Surface Area} = \pi \times r \times l \][/tex]
where [tex]\( r \)[/tex] is the radius of the cone and [tex]\( l \)[/tex] is the slant height.
1. Calculate the radius:
[tex]\[ r = \frac{\text{diameter}}{2} = \frac{3.5}{2} = 1.75 \text{ inches} \][/tex]
2. Calculate the slant height [tex]\( l \)[/tex]:
[tex]\[ l = \sqrt{r^2 + h^2} = \sqrt{1.75^2 + 8^2} = \sqrt{3.0625 + 64} = \sqrt{67.0625} = 8.19 \text{ inches} \][/tex]
3. Calculate the lateral surface area:
[tex]\[ \text{Lateral Surface Area} = \pi \times 1.75 \times 8.19 = 45.02 \text{ square inches} \][/tex]
4. Round to the nearest square inch:
[tex]\[ \text{Rounded Surface Area} = 45 \text{ square inches} \][/tex]
So, the waffle cone takes approximately [tex]\( 45 \)[/tex] square inches of waffle material to make.
#### Part b: Volume of the cone
To find the volume of a cone, we use the formula:
[tex]\[ \text{Volume} = \frac{1}{3} \pi r^2 h \][/tex]
1. Calculate the volume:
[tex]\[ \text{Volume} = \frac{1}{3} \pi (1.75)^2 \times 8 = \frac{1}{3} \pi \times 3.0625 \times 8 = \frac{1}{3} \pi \times 24.5 = 25.66 \text{ cubic inches} \][/tex]
2. Round to the nearest cubic inch:
[tex]\[ \text{Rounded Volume} = 26 \text{ cubic inches} \][/tex]
So, the cone can hold approximately [tex]\( 26 \)[/tex] cubic inches of ice cream.
#### Part c: Number of spherical scoops
Next, we need to find out how many perfectly spherical scoops of ice cream (with diameter 3.5 inches) can fit into the cone.
1. Calculate the volume of a spherical scoop:
[tex]\[ \text{Volume} = \frac{4}{3} \pi r^3 \][/tex]
Since the diameter of the scoop is 3.5 inches, the radius is:
[tex]\[ r = \frac{3.5}{2} = 1.75 \text{ inches} \][/tex]
[tex]\[ \text{Volume} = \frac{4}{3} \pi (1.75)^3 = \frac{4}{3} \pi \times 5.359375 = 22.45 \text{ cubic inches} \][/tex]
2. Calculate the number of scoops:
[tex]\[ \text{Number of scoops} = \frac{\text{Volume of cone}}{\text{Volume of one scoop}} = \frac{25.66}{22.45} = 1.14 \][/tex]
3. Round to the nearest whole number:
[tex]\[ \text{Rounded Number of Scoops} = 1 \][/tex]
So, approximately 1 spherical scoop of ice cream could melt into the cone without overflowing.
### Summary of Answers:
a. [tex]\( 45 \)[/tex] square inches
b. [tex]\( 26 \)[/tex] cubic inches
c. [tex]\( 1 \)[/tex] spherical scoop
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