Please help me solve: Determine surface areas of composite objects
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HallsofIvy
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This is a tedious problem but requires only very basic geometry formulas such as the area of a triangle and the surface area of a cylinder. Do you know those? There is a little "trick" involved in that we do not want to include the area of one end of the cyinder. In fact, we have to subtract that from the area of one of the triangular side.
This reduces to the surface area of a cylinder, counting only the bottom end, not the top. To find the area of the curved surface, imagine cutting down the length of the cylinder, the fold it out. It will be a rectangle with length equal to 2.5 m and width the circumference of a circle with radius 0.5 m. What is that circumference? So what is the area of the curved surface? The bottom is a disk with radius 0.5 m. What is its area?
The top figure can be thought of as 5 separate regions. Two are rectangles with length 2.5 m and width 1.0 m. One is a rectangle with length 3.0 m and width 1.0 m. What are the areas of those? The top is a rectangle with one side of length 3.0 m and two of length 2.5 m. The area of a triangle is "1/2 base times height". We can take the base to be the 3.0 m side. To find the height, imagine a line from the vertex where the two 2.5 m sides meet perpendicular to the 3.0 m side. That line also bisects the 3.0 m side so divides the triangle into two right triangles having leg of length 3.0/2= 1.5 m and hypotenuse of length 2.5 m. Use the Pythagorean theorem to determine the side of the other leg (the height of the original triangle. The bottom triangle is the same except that you will need to subtract the area of the top of the cylinder that covers that part of the triangle. It, like the bottom of the cylinder, is a disk with radius 0.5 m.
This reduces to the surface area of a cylinder, counting only the bottom end, not the top. To find the area of the curved surface, imagine cutting down the length of the cylinder, the fold it out. It will be a rectangle with length equal to 2.5 m and width the circumference of a circle with radius 0.5 m. What is that circumference? So what is the area of the curved surface? The bottom is a disk with radius 0.5 m. What is its area?
The top figure can be thought of as 5 separate regions. Two are rectangles with length 2.5 m and width 1.0 m. One is a rectangle with length 3.0 m and width 1.0 m. What are the areas of those? The top is a rectangle with one side of length 3.0 m and two of length 2.5 m. The area of a triangle is "1/2 base times height". We can take the base to be the 3.0 m side. To find the height, imagine a line from the vertex where the two 2.5 m sides meet perpendicular to the 3.0 m side. That line also bisects the 3.0 m side so divides the triangle into two right triangles having leg of length 3.0/2= 1.5 m and hypotenuse of length 2.5 m. Use the Pythagorean theorem to determine the side of the other leg (the height of the original triangle. The bottom triangle is the same except that you will need to subtract the area of the top of the cylinder that covers that part of the triangle. It, like the bottom of the cylinder, is a disk with radius 0.5 m.
Thanks for the reply. I manged to work out the problem and got 24.58 but the back of the book says 21.9. I have tried a million different ways and cannot get that answer.. or anything close to it. Perhaps the book is wrong.
For the sides of the triangle I have:
1x3 + 1x2.5 + 1x2.5 = 8
Surface of top triangle:
1.5 squared + 2.5 squared = c squared
2.25 + 6.25 = c squared
2.91 = c aka height
1/2 base x height
1.5 x 2.91 = 4.365
Total surface are for triangle is 8 + 4.365 + 4.365 = 16.73 (i still need to subtract out the cylinder top from below
Total triangle with cylinder top subtracted = 16.73 - 0.785 = 15.945
cylinder top is
pi r squared
3.14 x 0.5 squared
0.785
Cylinder body and base
2 pi r h + 2 pi r squared
2 x 3.14 x 0.5 x 2.5 + 2 (3.14x0.5 suared)
7.85 + 1.57
9.42 total (but I need to subtract the top of the cylinder)
9.42 - 0.785 = 8.635
Final answer 8.635 + 15.945 = 24.58
What am I doing wrong?? or is the book wrong.
Steven G
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I see two error. One is in red above. Th other is that you never included the top of the cylinder. That is you did not include it in the bottom of the triangle nor the cylinder. You need to include it once I would think.
The 'error' you have in red I believe is correct. This is taken from the first post
Also... I did include the base of the cylinder once. I subtracted it from the triangle surface area. But then when calculating the surface area of the cylinder the formula calculates both ends.. so once again I had to subtract the top one. Thus leaving the bottom one counted 1 time.
Steven G
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Given a right triangle and sides a, b and c it is NOT always true that a^2 + b^2 = c^2, It is only true if c is the hypotenuse of the triangle. I asked once before, what is the length of the hypotenuse in your right triangle?
Fine you included the base of the cylinder once like you should. But you never included the top even once. You removed it from the triangle region and from the top of the cylinder. It might be the case that there is a hole in the bottom of the triangular region and the cylinder without a top is glued into it but I think that you'd be reading too much into the problem.
Thanks for the help everyone. I see what you are saying. I basically did the Pythagorean theorem wrong 
For the sides of the triangle I have:
1x3 + 1x2.5 + 1x2.5 = 8
Surface of top triangle:
a squared + b squared = c squared (hypotenuse) (hypotenuse is always opposite the right angle.. this is the part I messed up. I already had this value and was trying to solve for it)
1.5 squared + b squared = 2.5 squared
2.25 + b squared = 6.25
b squared = 6.25 - 2.25
b squared = 4
b = 2
1/2 base x height
1.5 x 2 = 3
Total surface are for triangle is 8 + 3 + 3 = 14 (i still need to subtract out the cylinder top from below
Total triangle with cylinder top subtracted = 14 - 0.785 = 13.215
cylinder top is
pi r squared
3.14 x 0.5 squared
0.785
Cylinder body and base
2 pi r h + 2 pi r squared
2 x 3.14 x 0.5 x 2.5 + 2 (3.14x0.5 suared)
7.85 + 1.57
9.42 total (but I need to subtract the top of the cylinder)
9.42 - 0.785 = 8.635
Final answer 8.635 + 13.215 = 21.85 (rounded to 21.9 - which is the answer in the book
Thanks again everyone...
For the sides of the triangle I have:
1x3 + 1x2.5 + 1x2.5 = 8
Surface of top triangle:
a squared + b squared = c squared (hypotenuse) (hypotenuse is always opposite the right angle.. this is the part I messed up. I already had this value and was trying to solve for it)
1.5 squared + b squared = 2.5 squared
2.25 + b squared = 6.25
b squared = 6.25 - 2.25
b squared = 4
b = 2
1/2 base x height
1.5 x 2 = 3
Total surface are for triangle is 8 + 3 + 3 = 14 (i still need to subtract out the cylinder top from below
Total triangle with cylinder top subtracted = 14 - 0.785 = 13.215
cylinder top is
pi r squared
3.14 x 0.5 squared
0.785
Cylinder body and base
2 pi r h + 2 pi r squared
2 x 3.14 x 0.5 x 2.5 + 2 (3.14x0.5 suared)
7.85 + 1.57
9.42 total (but I need to subtract the top of the cylinder)
9.42 - 0.785 = 8.635
Final answer 8.635 + 13.215 = 21.85 (rounded to 21.9 - which is the answer in the book
Thanks again everyone...