When baking a cake, you have a choice of the following pans:
a round cake pan that is 2 inches deep and has a 7 inch diameter
a 6 inch × 9 inch rectangular cake pan that is 2 inches deep
a. Which of these pans has the larger volume? Justify your answer.
b. Excluding the bottom of the cake, what is the surface area of each cake that will be covered by
frosting? Justify your answer.
I just need help with b
Steve answered: volumes are
round: πr^2h = π*(7/2)^2*2 = 98π/4 = 76.969 in^3
parallelopiped: LWH = 6*9*2 = 108 in^3
should be easy to tell now, eh?
areas are
round: πr^2 + 2πrh = πr(r+2h)
brick: LW+2LH+2WH = LW+2H(L+W)
Now plug in your numbers and compare
Then I asked: So...
Area for round pan: 3.14*3.5*2+2*3.14*3.5*3.5=98.91
and...
Area for rectangular pan:
9*6+2*9*2+2*6*2=114
Am I right if not help me with b
Steve answered: no, you used 5*2=10 instead of 5^2=25 for the round area.
Then I asked: am I right now?
Area for round pan: 3.14*3.5^2+2*3.14*3.5*3.5=115.395
Area for rectangular pan:
9*6+2*9*2+2*6*2=114
Jeez Cazzo. You show up everywhere. Go harass someone who actually cares. I mean we`re only checking our answers.
@ Mrs. Verlarde I looked at your account, and all you do is post that same thing on every math thing you see. Come on! Seriously?
nevermind I think I got it
No, you are not correct. Let's calculate the surface area of each cake pan again.
For the round pan:
- The formula for the area of a circle is A = πr^2, where r is the radius.
- In this case, the radius is half of the diameter given, so r = 7/2 = 3.5 inches.
- The formula for the lateral surface area of a cylinder is A = 2πrh, where h is the height.
- In this case, the height of the round pan is given as 2 inches.
- So, the surface area of the round pan is A = πr^2 + 2πrh = π(3.5)^2 + 2π(3.5)(2).
For the rectangular pan:
- The formula for the area of a rectangle is A = LW, where L is the length and W is the width.
- In this case, the length is 9 inches and the width is 6 inches.
- The formula for the surface area of a rectangular prism is A = 2LW + 2LH + 2WH, where H is the height.
- In this case, the height of the rectangular pan is given as 2 inches.
- So, the surface area of the rectangular pan is A = LW + 2LH + 2WH = (9)(6) + 2(9)(2) + 2(6)(2).
Now, let's plug in the values and calculate the surface area for each pan.
For the round pan:
A = π(3.5)^2 + 2π(3.5)(2)
A ≈ 3.14 × 12.25 + 2 × 3.14 × 3.5 × 2
A ≈ 38.465 + 43.96
A ≈ 82.425 square inches
For the rectangular pan:
A = (9)(6) + 2(9)(2) + 2(6)(2)
A = 54 + 36 + 24
A = 114 square inches
So, the correct surface areas are:
- Round pan: 82.425 square inches
- Rectangular pan: 114 square inches
Therefore, the surface area covered by frosting would be greater for the rectangular pan, with a surface area of 114 square inches.