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Sagot :
To solve this problem, we need to determine the amount of icing required for both Lauren and Patrick's cakes and then compare the two.
### Step-by-Step Solution
Given:
- Volume of sponge each uses:
[tex]\[ \text{sponge volume} = \frac{12,000 \pi}{3} \, \text{cm}^3 \][/tex]
For Lauren:
1. Volume of each cake:
[tex]\[ V_{\text{Lauren}} = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi \ (5)^3 = \frac{4}{3} \pi \ (125) = \frac{500 \pi}{3} \, \text{cm}^3 \][/tex]
2. Number of cakes Lauren can make:
[tex]\[ \text{num cakes}_{\text{Lauren}} = \frac{\text{sponge volume}}{V_{\text{Lauren}}} = \frac{\frac{12,000 \pi}{3}}{\frac{500 \pi}{3}} = \frac{12,000 \pi /3}{500 \pi /3} = \frac{12,000}{500} = 24 \][/tex]
3. Surface area of each cake:
[tex]\[ SA_{\text{Lauren}} = 4 \pi r^2 = 4 \pi (5)^2 = 4 \pi (25) = 100 \pi \, \text{cm}^2 \][/tex]
4. Total surface area Lauren needs to cover:
[tex]\[ \text{total SA}_{\text{Lauren}} = \text{num cakes}_{\text{Lauren}} \times SA_{\text{Lauren}} = 24 \times 100 \pi = 2400 \pi \, \text{cm}^2 \][/tex]
For Patrick:
1. Volume of each cake:
[tex]\[ V_{\text{Patrick}} = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi \ (2)^3 = \frac{4}{3} \pi \ (8) = \frac{32 \pi}{3} \, \text{cm}^3 \][/tex]
2. Number of cakes Patrick can make:
[tex]\[ \text{num cakes}_{\text{Patrick}} = \frac{\text{sponge volume}}{V_{\text{Patrick}}} = \frac{\frac{12,000 \pi}{3}}{\frac{32 \pi}{3}} = \frac{12,000 \pi /3}{32 \pi /3} = \frac{12,000}{32} = 375 \][/tex]
3. Surface area of each cake:
[tex]\[ SA_{\text{Patrick}} = 4 \pi r^2 = 4 \pi (2)^2 = 4 \pi (4) = 16 \pi \, \text{cm}^2 \][/tex]
4. Total surface area Patrick needs to cover:
[tex]\[ \text{total SA}_{\text{Patrick}} = \text{num cakes}_{\text{Patrick}} \times SA_{\text{Patrick}} = 375 \times 16 \pi = 6000 \pi \, \text{cm}^2 \][/tex]
Comparison:
Now we compare the total surface areas to determine who needs more icing:
[tex]\[ 2400 \pi \, \text{cm}^2 \, (\text{Lauren's total surface area}) \quad \text{and} \quad 6000 \pi \, \text{cm}^2 \, (\text{Patrick's total surface area}) \][/tex]
Clearly, Patrick needs more icing.
### Answer to Part (a):
Patrick needs more icing to cover all of their cakes.
### Answer to Part (b):
To find how much more area Patrick needs to cover compared to Lauren, we subtract Lauren's total surface area from Patrick's total surface area:
[tex]\[ 6000 \pi \, \text{cm}^2 - 2400 \pi \, \text{cm}^2 = 3600 \pi \, \text{cm}^2 \][/tex]
So, Patrick needs to cover [tex]\(3600 \pi \, \text{cm}^2\)[/tex] more than Lauren.
### Final Answer:
- a) Patrick needs more icing to cover all of their cakes.
- b) Patrick needs to cover [tex]\(3600 \pi \, \text{cm}^2\)[/tex] more area than Lauren.
### Step-by-Step Solution
Given:
- Volume of sponge each uses:
[tex]\[ \text{sponge volume} = \frac{12,000 \pi}{3} \, \text{cm}^3 \][/tex]
For Lauren:
1. Volume of each cake:
[tex]\[ V_{\text{Lauren}} = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi \ (5)^3 = \frac{4}{3} \pi \ (125) = \frac{500 \pi}{3} \, \text{cm}^3 \][/tex]
2. Number of cakes Lauren can make:
[tex]\[ \text{num cakes}_{\text{Lauren}} = \frac{\text{sponge volume}}{V_{\text{Lauren}}} = \frac{\frac{12,000 \pi}{3}}{\frac{500 \pi}{3}} = \frac{12,000 \pi /3}{500 \pi /3} = \frac{12,000}{500} = 24 \][/tex]
3. Surface area of each cake:
[tex]\[ SA_{\text{Lauren}} = 4 \pi r^2 = 4 \pi (5)^2 = 4 \pi (25) = 100 \pi \, \text{cm}^2 \][/tex]
4. Total surface area Lauren needs to cover:
[tex]\[ \text{total SA}_{\text{Lauren}} = \text{num cakes}_{\text{Lauren}} \times SA_{\text{Lauren}} = 24 \times 100 \pi = 2400 \pi \, \text{cm}^2 \][/tex]
For Patrick:
1. Volume of each cake:
[tex]\[ V_{\text{Patrick}} = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi \ (2)^3 = \frac{4}{3} \pi \ (8) = \frac{32 \pi}{3} \, \text{cm}^3 \][/tex]
2. Number of cakes Patrick can make:
[tex]\[ \text{num cakes}_{\text{Patrick}} = \frac{\text{sponge volume}}{V_{\text{Patrick}}} = \frac{\frac{12,000 \pi}{3}}{\frac{32 \pi}{3}} = \frac{12,000 \pi /3}{32 \pi /3} = \frac{12,000}{32} = 375 \][/tex]
3. Surface area of each cake:
[tex]\[ SA_{\text{Patrick}} = 4 \pi r^2 = 4 \pi (2)^2 = 4 \pi (4) = 16 \pi \, \text{cm}^2 \][/tex]
4. Total surface area Patrick needs to cover:
[tex]\[ \text{total SA}_{\text{Patrick}} = \text{num cakes}_{\text{Patrick}} \times SA_{\text{Patrick}} = 375 \times 16 \pi = 6000 \pi \, \text{cm}^2 \][/tex]
Comparison:
Now we compare the total surface areas to determine who needs more icing:
[tex]\[ 2400 \pi \, \text{cm}^2 \, (\text{Lauren's total surface area}) \quad \text{and} \quad 6000 \pi \, \text{cm}^2 \, (\text{Patrick's total surface area}) \][/tex]
Clearly, Patrick needs more icing.
### Answer to Part (a):
Patrick needs more icing to cover all of their cakes.
### Answer to Part (b):
To find how much more area Patrick needs to cover compared to Lauren, we subtract Lauren's total surface area from Patrick's total surface area:
[tex]\[ 6000 \pi \, \text{cm}^2 - 2400 \pi \, \text{cm}^2 = 3600 \pi \, \text{cm}^2 \][/tex]
So, Patrick needs to cover [tex]\(3600 \pi \, \text{cm}^2\)[/tex] more than Lauren.
### Final Answer:
- a) Patrick needs more icing to cover all of their cakes.
- b) Patrick needs to cover [tex]\(3600 \pi \, \text{cm}^2\)[/tex] more area than Lauren.
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