SOLUTION: You own a bakery which specializes in custom orders. A customer requests to have a three-tiered cake made. Requirements are as follows: Each tier is 15 cm high; The bot

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Question 1043151: You own a bakery which specializes in custom orders. A customer requests to have a three-tiered cake made. Requirements are as follows:
Each tier is 15 cm high;
The bottom tier is a cylinder with a diameter of 45 cm;
The middle tier is a square–based prism. It must be positioned so that its corners touch the circumference of the bottom tier;
The top tier is another cylinder inscribed in the middle tier.
You know that 250 mL of raw batter will produce 625 mL of cake. How much raw batter is required for this cake? Note: 1 mL = 1 cm3
Thanks so much :)
Found 3 solutions by ikleyn, josgarithmetic, MathTherapy:
Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.
You own a bakery which specializes in custom orders. A customer requests to have a three-tiered cake made. Requirements are as follows:
Each tier is 15 cm high;
The bottom tier is a cylinder with a diameter of 45 cm;
The middle tier is a square–based prism. It must be positioned so that its corners touch the circumference of the bottom tier;
The top tier is another cylinder inscribed in the middle tier.
You know that 250 mL of raw batter will produce 625 mL of cake. How much raw batter is required for this cake? Note: 1 mL = 1 cm3
Thanks so much :)
~~~~~~~~~~~~~~~~~~~~~~~~~ 

1.  The volume of the bottom tier is  pi%2A%28D%2F2%29%5E2%2Ah = 3.14%2A%2845%2F2%29%5E2%2A15 = 23844.4 cm%5E3.

2.  The square side of the middle tier is  a = 45%2Fsqrt%282%29 cm.

    Then the volume of the middle tier is  a%5E2%2Ah = %2845%2Fsqrt%282%29%29%5E2%2A15 = 15187.5 cm%5E3.

3.  The diameter of the top tier is  d = a = 45%2Fsqrt%282%29.

    Then the volume of the top tier is  pi%2A%28d%2F2%29%5E2%2Ah = 3.14%2A%2845%2F%282%2Asqrt%282%29%29%29%5E2%2A15 =  11922.2 cm%5E3.   (notice it is half the volume of the top tier !)

4. The sum of volumes 1, 2, and 3 is  23844.4 + 15187.5 + 11922.2 = 50954 cm%5E3.

5. 50954%2F625 = 81.5 times.

6. Last step is to multiply  250 mL of raw batter by 81.5.

   It gives 250*81.5 = 20381.6 mL = 20.3816 L of raw batter.

Answer.  20381.6 mL = 20.3816 L of raw batter.


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comment from student: I am not familiar with this>>>The square side of the middle tier you have
as a= 45 divided by the square root of 2, i have never done it that way or seen it done that way,
as i am in grade 12 math. What is the a^2 stand for? Thanks :)
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My response: If a square is inscribed into a circle of the diameter d, then the side of the square is d%2Fsqrt%282%29.
Which is the same as %28d%2Asqrt%282%29%29%2F2.

It is so elementary . . .

Next your question is: "What is the a^2 stand for?"

a%5E2 stands for a*a.

I also have a question. Say, is it true that you are in grade 12 math?


Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
First goal is find the volume of each tier.
Second goal is sum the three volumes.
Third goal is find needed volume of raw batter.

BOTTOM
volume is 15%2Api%2A%2822.5%29%5E2 cubic cm's.

MIDDLE
You need to find the square side length of an edge.
A diagonal of square side of the prism is diameter of the bottom tier.
If this diagonal is understood to be cutting two right isosceles from the square lower side or top side, then for side s,
2s%5E2=45%5E2
s%5E2=45%5E2%2F2
s=45%2Fsqrt%282%29
s=45sqrt%282%29%2F2
and volume of this tier is s%5E3;---------- ** (No. See below.)
or
%2845%2F2%29%5E3%2Asqrt%282%29%28sqrt%282%29%29%5E2
%2845%2A45%2A45%2Asqrt%282%29%2A2%29%2F%282%2A4%29
45%5E3%2Asqrt%282%29%2F4 ---------------------** (see below)
-
Correcting this middle tier volume right now.
Volume here will be 15%2As%2As, because each tier is given 15 cm tall.
This volume,.



TOP TIER
A cylinder, with diameter being same as edge length of the tier below it.
This edge length found was s=45sqrt%282%29%2F2.
Volume, 15%2Api%2A%28s%2F2%29%5E2, because we can use RADIUS instead of the full diameter...
15pi%2A%2845sqrt%282%29%2F2%29%5E2----*
15%2A45%5E2%2Api%2A2%2F4
15%2A45%5E2%2Api%2F2
-
Correcting this part...
Diameter s=45sqrt%282%29%2F2
RADIUS s%2F2=%2845sqrt%282%29%2F2%29%2F2
RADIUS, %2845%2F4%29sqrt%282%29
Now find this top tier volume. Cylinder.
15%2Api%28%2845%2F4%29sqrt%282%29%29%5E2
%28%2815%2Api%2A45%5E2%29%2F4%5E2%29%2A2
highlight_green%28pi%2815%2A45%5E2%29%2F2%5E3%29

FINAL CAKE VOLUME
sum of the finished three tiers, cubic centimeters
-----(No. Still Wrong. Not yet corrected)
.
.
.
but this is not finished solving. This is how much CAKE, but not yet how much batter. Still is needed, simplification, and then using the ratio for volumes of batter to cake of 250%2F625, and any other computations to finish with a final value answer.

--
* This appears to be a mistake. The correct expression before computation should be somewhat different...
-
** Another mistake found. Forgot, each tier is 15 cm tall. This tier is not simply s%5E3.
-
Corrections are now shown for tier middle and tier top, but corrections not yet made for CAKE VOLUME.

Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!
You own a bakery which specializes in custom orders. A customer requests to have a three-tiered cake made. Requirements are as follows:
Each tier is 15 cm high;
The bottom tier is a cylinder with a diameter of 45 cm;
The middle tier is a square–based prism. It must be positioned so that its corners touch the circumference of the bottom tier;
The top tier is another cylinder inscribed in the middle tier.
You know that 250 mL of raw batter will produce 625 mL of cake. How much raw batter is required for this cake? Note: 1 mL = 1 cm3
Thanks so much :)
Volume of bottom tier (cylinder): matrix%281%2C4%2C+%227%2C593.75%22%2C+pi%2C+cu%2C+cm%29

Volume of middle tier (square prism): matrix%281%2C3%2C+%2215%2C187.5%22%2C+cu%2C+cm%29

Volume of top tier (cylinder): matrix%281%2C4%2C+%223%2C796.875%22%2C+pi%2C+cu%2C+cm%29

Volume of cake: 



Since 250 mL of raw butter will produce 625 mL of cake, and with amount of raw butter needed, being B, we get: 250%2FB+=+625%2F%2250%2C972.2%22

625B+=+250%28%2250%2C972.2%22%29 ------- Cross-multiplying

B, or amount of raw butter needed =