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Sagot :
We are given a cylindrical ice cream tub with a total surface area of 183.69 square inches and a diameter of 6 inches. We need to find the height of the tub. We are also given that [tex]\(\pi = 3.14\)[/tex].
Here is the step-by-step solution to determine the height:
1. Find the radius of the tub:
The radius [tex]\(r\)[/tex] is half of the diameter.
[tex]\[ r = \frac{\text{diameter}}{2} = \frac{6}{2} = 3 \text{ inches} \][/tex]
2. Surface area formula for a cylinder:
The total surface area (SA) of a cylinder is given by:
[tex]\[ SA = 2\pi rh + 2\pi r^2 \][/tex]
where [tex]\(r\)[/tex] is the radius and [tex]\(h\)[/tex] is the height.
3. Rearrange the formula to solve for height [tex]\(h\)[/tex]:
[tex]\[ SA = 2\pi rh + 2\pi r^2 \][/tex]
Subtract [tex]\(2\pi r^2\)[/tex] from both sides:
[tex]\[ SA - 2\pi r^2 = 2\pi rh \][/tex]
Divide both sides by [tex]\(2\pi r\)[/tex]:
[tex]\[ h = \frac{SA - 2\pi r^2}{2\pi r} \][/tex]
4. Calculate the area of the two end circles:
The area of one end circle is [tex]\(\pi r^2\)[/tex], so for two end circles:
[tex]\[ 2 \pi r^2 = 2 \times 3.14 \times 3^2 = 2 \times 3.14 \times 9 = 56.52 \text{ square inches} \][/tex]
5. Calculate the height [tex]\(h\)[/tex]:
Substitute the values [tex]\(SA = 183.69\)[/tex], [tex]\(\pi = 3.14\)[/tex], and [tex]\(r = 3\)[/tex] into the rearranged formula:
[tex]\[ h = \frac{183.69 - 56.52}{2 \times 3.14 \times 3} \][/tex]
Simplify the numerator:
[tex]\[ 183.69 - 56.52 = 127.17 \][/tex]
Now, calculate the denominator:
[tex]\[ 2 \times 3.14 \times 3 = 6.28 \times 3 = 18.84 \][/tex]
Finally, divide the numerator by the denominator:
[tex]\[ h = \frac{127.17}{18.84} \approx 6.75 \text{ inches} \][/tex]
Hence, the height of the tub is approximately [tex]\(6.75\)[/tex] inches.
Therefore, the correct answer is:
[tex]\[ \boxed{6.75 \text{ inches}} \][/tex]
Here is the step-by-step solution to determine the height:
1. Find the radius of the tub:
The radius [tex]\(r\)[/tex] is half of the diameter.
[tex]\[ r = \frac{\text{diameter}}{2} = \frac{6}{2} = 3 \text{ inches} \][/tex]
2. Surface area formula for a cylinder:
The total surface area (SA) of a cylinder is given by:
[tex]\[ SA = 2\pi rh + 2\pi r^2 \][/tex]
where [tex]\(r\)[/tex] is the radius and [tex]\(h\)[/tex] is the height.
3. Rearrange the formula to solve for height [tex]\(h\)[/tex]:
[tex]\[ SA = 2\pi rh + 2\pi r^2 \][/tex]
Subtract [tex]\(2\pi r^2\)[/tex] from both sides:
[tex]\[ SA - 2\pi r^2 = 2\pi rh \][/tex]
Divide both sides by [tex]\(2\pi r\)[/tex]:
[tex]\[ h = \frac{SA - 2\pi r^2}{2\pi r} \][/tex]
4. Calculate the area of the two end circles:
The area of one end circle is [tex]\(\pi r^2\)[/tex], so for two end circles:
[tex]\[ 2 \pi r^2 = 2 \times 3.14 \times 3^2 = 2 \times 3.14 \times 9 = 56.52 \text{ square inches} \][/tex]
5. Calculate the height [tex]\(h\)[/tex]:
Substitute the values [tex]\(SA = 183.69\)[/tex], [tex]\(\pi = 3.14\)[/tex], and [tex]\(r = 3\)[/tex] into the rearranged formula:
[tex]\[ h = \frac{183.69 - 56.52}{2 \times 3.14 \times 3} \][/tex]
Simplify the numerator:
[tex]\[ 183.69 - 56.52 = 127.17 \][/tex]
Now, calculate the denominator:
[tex]\[ 2 \times 3.14 \times 3 = 6.28 \times 3 = 18.84 \][/tex]
Finally, divide the numerator by the denominator:
[tex]\[ h = \frac{127.17}{18.84} \approx 6.75 \text{ inches} \][/tex]
Hence, the height of the tub is approximately [tex]\(6.75\)[/tex] inches.
Therefore, the correct answer is:
[tex]\[ \boxed{6.75 \text{ inches}} \][/tex]
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