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Sagot :
Let's first identify the problem and then work through each step carefully.
### Part (a): Who needs more icing to cover all of their cakes?
1. Volume of Sponge Each Used:
Each of them uses \(\frac{8000 \pi}{3} \, \text{cm}^3\) of sponge.
2. Volume of Tanya’s Cake:
The volume \(V\) of a sphere (cake) is given by \(\frac{4}{3} \pi r^3\).
For Tanya, \( r = 5 \) cm.
[tex]\[ V_{\text{Tanya}} = \frac{4}{3} \pi (5^3) = \frac{4}{3} \pi (125) = \frac{500 \pi}{3} \, \text{cm}^3. \][/tex]
3. Volume of Russel’s Cake:
For Russel, \( r = 2 \) cm.
[tex]\[ V_{\text{Russel}} = \frac{4}{3} \pi (2^3) = \frac{4}{3} \pi (8) = \frac{32 \pi}{3} \, \text{cm}^3. \][/tex]
4. Number of Cakes Each can Make:
Using the total volume of sponge, calculate how many cakes they can each make.
[tex]\[ \text{Cakes Tanya} = \frac{\frac{8000 \pi}{3}}{\frac{500 \pi}{3}} = \frac{8000}{500} = 16 \][/tex]
[tex]\[ \text{Cakes Russel} = \frac{\frac{8000 \pi}{3}}{\frac{32 \pi}{3}} = \frac{8000}{32} = 250 \][/tex]
5. Surface Area of Each Cake:
The surface area \(A\) of a sphere is given by \(4 \pi r^2\).
For Tanya, \( r = 5 \) cm.
[tex]\[ A_{\text{Tanya}} = 4 \pi (5^2) = 4 \pi (25) = 100 \pi \, \text{cm}^2. \][/tex]
For Russel, \( r = 2 \) cm.
[tex]\[ A_{\text{Russel}} = 4 \pi (2^2) = 4 \pi (4) = 16 \pi \, \text{cm}^2. \][/tex]
6. Total Surface Area to Cover with Icing:
Multiply the number of cakes by the surface area of each cake.
[tex]\[ \text{Total Surface Area Tanya} = 16 \times 100 \pi = 1600 \pi \, \text{cm}^2. \][/tex]
[tex]\[ \text{Total Surface Area Russel} = 250 \times 16 \pi = 4000 \pi \, \text{cm}^2. \][/tex]
Comparing the total surface areas, we can see that Russel needs more icing because:
[tex]\[ 4000 \pi > 1600 \pi. \][/tex]
### Part (b): How Much More Area Does This Person Need to Cover?
To find the additional area Russel needs to cover compared to Tanya, we subtract the total surface area needed by Tanya from the total surface area needed by Russel.
[tex]\[ \text{Area Difference} = 4000 \pi \, \text{cm}^2 - 1600 \pi \, \text{cm}^2 = 2400 \pi \, \text{cm}^2. \][/tex]
### Conclusion:
(a) Russel needs more icing to cover all of his cakes.
(b) Russel needs to cover an additional [tex]\(2400 \pi\)[/tex] cm[tex]\(^2\)[/tex].
### Part (a): Who needs more icing to cover all of their cakes?
1. Volume of Sponge Each Used:
Each of them uses \(\frac{8000 \pi}{3} \, \text{cm}^3\) of sponge.
2. Volume of Tanya’s Cake:
The volume \(V\) of a sphere (cake) is given by \(\frac{4}{3} \pi r^3\).
For Tanya, \( r = 5 \) cm.
[tex]\[ V_{\text{Tanya}} = \frac{4}{3} \pi (5^3) = \frac{4}{3} \pi (125) = \frac{500 \pi}{3} \, \text{cm}^3. \][/tex]
3. Volume of Russel’s Cake:
For Russel, \( r = 2 \) cm.
[tex]\[ V_{\text{Russel}} = \frac{4}{3} \pi (2^3) = \frac{4}{3} \pi (8) = \frac{32 \pi}{3} \, \text{cm}^3. \][/tex]
4. Number of Cakes Each can Make:
Using the total volume of sponge, calculate how many cakes they can each make.
[tex]\[ \text{Cakes Tanya} = \frac{\frac{8000 \pi}{3}}{\frac{500 \pi}{3}} = \frac{8000}{500} = 16 \][/tex]
[tex]\[ \text{Cakes Russel} = \frac{\frac{8000 \pi}{3}}{\frac{32 \pi}{3}} = \frac{8000}{32} = 250 \][/tex]
5. Surface Area of Each Cake:
The surface area \(A\) of a sphere is given by \(4 \pi r^2\).
For Tanya, \( r = 5 \) cm.
[tex]\[ A_{\text{Tanya}} = 4 \pi (5^2) = 4 \pi (25) = 100 \pi \, \text{cm}^2. \][/tex]
For Russel, \( r = 2 \) cm.
[tex]\[ A_{\text{Russel}} = 4 \pi (2^2) = 4 \pi (4) = 16 \pi \, \text{cm}^2. \][/tex]
6. Total Surface Area to Cover with Icing:
Multiply the number of cakes by the surface area of each cake.
[tex]\[ \text{Total Surface Area Tanya} = 16 \times 100 \pi = 1600 \pi \, \text{cm}^2. \][/tex]
[tex]\[ \text{Total Surface Area Russel} = 250 \times 16 \pi = 4000 \pi \, \text{cm}^2. \][/tex]
Comparing the total surface areas, we can see that Russel needs more icing because:
[tex]\[ 4000 \pi > 1600 \pi. \][/tex]
### Part (b): How Much More Area Does This Person Need to Cover?
To find the additional area Russel needs to cover compared to Tanya, we subtract the total surface area needed by Tanya from the total surface area needed by Russel.
[tex]\[ \text{Area Difference} = 4000 \pi \, \text{cm}^2 - 1600 \pi \, \text{cm}^2 = 2400 \pi \, \text{cm}^2. \][/tex]
### Conclusion:
(a) Russel needs more icing to cover all of his cakes.
(b) Russel needs to cover an additional [tex]\(2400 \pi\)[/tex] cm[tex]\(^2\)[/tex].
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