Surface area of 3D shape: rect. block w/ many half-cylinders removed

[Gojam123]

I've been trying to get this question done for the past few days, and I can't come up with any way to check whether it's correct or whether I'm missing something. I'm pretty sure the textbook's answer is wrong (not the first time), but I can't find any way to confirm my answer.

My working is as follows:

Surface area of entire shape = (110 x 45) + (110 x 35) + (45 x 35) x 2
= 20750cm²

20750 - (7.5 x 7.5 x 3.14159 x 4)
Note: This is for the 8 semi-circles. 8 semi-circles are equal to 4 normal circles, hence the x4.
= 20043.1

20043.1 - (15 x 35 x 4)
Note: This is accounting for the four rectangles.
= 17943.1

17943.1 + (15 x 3.14159 x 0.5 x 35 x 4)
Note: This is accounting for the new curved areas.
= 21241.8


Another way I solved it:

My reasoning is as follows:
Area of all semicircles = 706.9
Area of rectangular areas = 2100
Area of curved areas = 3298.7

3298.7 - 706.9 - 2100 = 491.8

Therefore, the answer has to be more than 20750. Moreover, it has to be 491.8 more than 20750, which is 21241.8, or my answer.
The textbook's answer is 19757

I can't get my teacher to check it because school just ended for the year.

 

[HallsofIvy]

I would start by finding the surface area of the rectangular solid. If I am reading this correctly (that "20" in bold at the top concerns me but I assume it is a length like all the others) then the rectangular solid is 10+ 15+ 20= 45 in one dimension, 35 in another, and 10+ 15+ 60+ 15+ 10= 110 in the third so would have surface area 2(45)(35)+ 2(45)(110)+ 2(35)(110)= 20750.

Now subtract the total of all rectangular areas that were cut off that surface by the cylindrical cutouts: all four are 15 by 35 so we must subtract 4(15)(35)= 2100. 20750- 2100= 18650.

Finally, add on the surface areas of the four semi-cylindrical cutouts. Since all four are the same that is the same as two cylinders with radius 15/2= 7.5 and length 35. That area is 2(2 pi)(7.5)(35)= 3299 (to the nearest whole number).

The total surface area is 18650+ 3299= 21949.

 

[Gojam123]

If I am reading this correctly (that "20" in bold at the top concerns me but I assume it is a length like all the others).

Yes, you're correct. I just did a quick edit to the picture I took as the lighting made that number invisible.

 

[Gojam123]

Thanks a lot for your reply! I can finally move on without the guilty conscience of having skipped a question.