1
00:00:00,000 --> 00:00:00,530


2
00:00:00,530 --> 00:00:05,110
Find the volume of a tank whose base has an area of 3x

3
00:00:05,110 --> 00:00:08,859
squared plus 30x plus 5 square feet and whose

4
00:00:08,859 --> 00:00:10,689
height is 8x minus 5.

5
00:00:10,689 --> 00:00:13,879
So let me draw a potential tank here.

6
00:00:13,880 --> 00:00:17,150
Maybe it's a cylinder, so let me draw a cylindrical tank

7
00:00:17,149 --> 00:00:21,159
here, just like that.

8
00:00:21,160 --> 00:00:23,260
It's supposed to be a cylinder, and maybe if it was

9
00:00:23,260 --> 00:00:25,860
transparent, you would see this back side over there.

10
00:00:25,859 --> 00:00:26,980
And these two are the same.

11
00:00:26,980 --> 00:00:28,850
These are supposed to be straight lines.

12
00:00:28,850 --> 00:00:32,658
And the volume of a-- let me just label it first. So the

13
00:00:32,658 --> 00:00:35,129
area of the base, which is the same as the area of the top

14
00:00:35,130 --> 00:00:40,690
here or of the base here, the area of that is 3x squared

15
00:00:40,689 --> 00:00:43,329
plus 30x plus 5.

16
00:00:43,329 --> 00:00:46,920
And they tell us the height is 8x minus 5.

17
00:00:46,920 --> 00:00:50,600
The height of this tank is 8x minus 5.

18
00:00:50,600 --> 00:00:54,890


19
00:00:54,890 --> 00:00:59,079
And if you want to find the volume of a three-dimensional

20
00:00:59,079 --> 00:01:01,539
object like this, you just multiply the area of the base

21
00:01:01,539 --> 00:01:02,719
times the height.

22
00:01:02,719 --> 00:01:07,349
So the volume is going to be the area of the base, which is

23
00:01:07,349 --> 00:01:13,500
3x squared plus 30x plus 5, times the height,

24
00:01:13,500 --> 00:01:15,439
times 8x minus 5.

25
00:01:15,439 --> 00:01:18,750


26
00:01:18,750 --> 00:01:22,030
Now, to multiply something like this out, it might seem

27
00:01:22,030 --> 00:01:24,549
really complicated, but we once again, we just have to do

28
00:01:24,549 --> 00:01:26,069
the distributive property.

29
00:01:26,069 --> 00:01:28,719
If you viewed this big thing in pink here as just a number,

30
00:01:28,719 --> 00:01:31,890
if this was like the number 7, you'd just say, well, this is

31
00:01:31,890 --> 00:01:37,430
going to be 7 times 8x minus 7 times 5 or minus 5 times 7.

32
00:01:37,430 --> 00:01:38,580
You would just distribute it out.

33
00:01:38,579 --> 00:01:42,399
You would just multiply this entire thing times each of the

34
00:01:42,400 --> 00:01:44,510
terms. That's what the distributive property tells us

35
00:01:44,510 --> 00:01:45,480
when you first learned it.

36
00:01:45,480 --> 00:01:46,410
So let's do that.

37
00:01:46,409 --> 00:01:49,939
So it's going to be this entire thing times 8x, or we

38
00:01:49,939 --> 00:01:52,439
can view it as 8x times this entire thing.

39
00:01:52,439 --> 00:02:05,599
So 8x times this entire thing: 3x squared plus 30x plus 5,

40
00:02:05,599 --> 00:02:10,978
minus 5 times the entire thing again, or the entire thing

41
00:02:10,979 --> 00:02:12,670
times minus 5.

42
00:02:12,669 --> 00:02:19,359
So once again you get 3x squared plus 30x plus 5.

43
00:02:19,360 --> 00:02:20,840
And now we just multiply these out.

44
00:02:20,840 --> 00:02:23,409
We just distribute out the 8x over this whole thing and we

45
00:02:23,409 --> 00:02:26,939
distribute out the negative 5 over that whole thing.

46
00:02:26,939 --> 00:02:32,050
So then we get 8x times 3x squared is 24x

47
00:02:32,050 --> 00:02:33,570
to the third power.

48
00:02:33,569 --> 00:02:35,569
8x times 30x is what?

49
00:02:35,569 --> 00:02:41,060
That's 240x squared, so plus 240x squared.

50
00:02:41,060 --> 00:02:45,620
8x times 5 is plus 40x.

51
00:02:45,620 --> 00:02:48,490
And then we multiply this negative 5 out.

52
00:02:48,490 --> 00:02:57,020
Negative 5 times 3x squared is negative 15x squared.

53
00:02:57,020 --> 00:03:02,030
Negative 5 times 30x is negative 150x.

54
00:03:02,030 --> 00:03:11,460
And then negative 5 times 5 is negative 25.

55
00:03:11,460 --> 00:03:13,450
And then we just have to simplify it from here.

56
00:03:13,449 --> 00:03:16,219
So we only have one third-degree term, one thing

57
00:03:16,219 --> 00:03:17,550
that has an x to the third in it.

58
00:03:17,550 --> 00:03:19,780
We have this term right here, so we'll write that

59
00:03:19,780 --> 00:03:21,810
as 24x to the third.

60
00:03:21,810 --> 00:03:23,949
And then what are our x squared terms?

61
00:03:23,949 --> 00:03:28,509
We have 240x squared minus 15x squared.

62
00:03:28,509 --> 00:03:31,019
So what's two 240 minus 15?

63
00:03:31,020 --> 00:03:32,960
It's 225x squared.

64
00:03:32,960 --> 00:03:36,240
So plus 225x squared.

65
00:03:36,240 --> 00:03:39,939
That's adding that term to that term right over there.

66
00:03:39,939 --> 00:03:47,099
And then we have 40x minus 150x.

67
00:03:47,099 --> 00:03:52,299
That would be negative 110x.

68
00:03:52,300 --> 00:03:57,330
And then finally we have just this negative 25 out here.

69
00:03:57,330 --> 00:03:59,240
That's the only constant term.

70
00:03:59,240 --> 00:04:00,240
And we're done!

71
00:04:00,240 --> 00:04:02,420
We found the volume of the tank.

72
00:04:02,419 --> 00:04:06,009
It's given by this polynomial expression right here.

73
00:04:06,009 --> 00:04:10,539
So this right here is the volume of the tank.

74
00:04:10,539 --> 00:04:15,090
It's equal to 24x to the third plus 225x squared

75
00:04:15,090 --> 00:04:18,680
minus 110x minus 25.