Question 1181985
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a baking pan has an rectangular base 12 in by 8 in, the sides and ends of the pan 
slope outward, so that the upper edges measure respectively 13 1/2 in by 9 in. 
if the depth of the pan is 2 in, find the amount of cake batter requires to fill 
the pan to 1/2 its depth
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<pre>
The base dimensions are 12 in by 8 in.


The upper edges measures are  13.5 inches  by  9 inches.


Dimensions at the 1/2 of the depth are (obviously)  12 + {{{(13.5-12)/2}}} = 12.75 inches  by  8 + {{{(9-8)/2}}} = 8.5 inches.


It is not difficult to check that the shape under consideration is a pyramidal frustum.


Apply the formula for the volume of a pyramidal frustum 


     V = {{{(h/3)*(S[t] + S[b] + sqrt(S[t]*S[b]))}}}.


Here h is the height of the frustum; {{{S[t]}}} is the area of the top base face = 12.75*8.5 = 108.375 in^2;

{{{S[b]}}} is the area of the bottom base face = 12*8 = 96 in^2.


So, the volume is


    V = {{{(1/3)*(108.375 + 96 + sqrt(108.375*96))}}} = 102.125 cubic inches.    <U>ANSWER</U>
</pre>

Solved.


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On the pyramidal frustum volume formula see the link


<A HREF=https://mathworld.wolfram.com/PyramidalFrustum.html>https://mathworld.wolfram.com/PyramidalFrustum.html</A>


https://mathworld.wolfram.com/PyramidalFrustum.html



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My answer differs from the answer by tutor @greenestamps.


It is because one formula in the solution by @greenestamps needs to be corrected.


This formula is for the volume of four pyramids  in section (3) of the solution by @greenestamps


    {{{(4)((1/3)((1/2)((1/4)h)((3/8)h)))(h) = (1/16)h^3}}}.


Its correct form is


    {{{(4)((1/3)(((1/4)h)((3/8)h)))(h) = (1/8)h^3}}}


without the factor {{{1/2}}}.



With this correction, the solution by @greenestamps gives the same final answer of 102.125 in^3, as my solution.