Question

6. (a) A square pyramid crystal has a square base with sides of length a and the height of the top vertex from the base given by h. The volume of the pyramid crystal is given by V = a^2 h / 3. [diagram] The crystal is growing in volume at a constant rate of 0.5 cm^3 per hour. At a particular point in time the crystal has height h = 1 cm and side a = ?2 cm. This ratio of height to side is constant as the crystal grows. How fast is the height increasing at this particular point in time? (b) Use the results from Section 3.4.1 (page 79) of your calculus notes (together with the properties of the ln(x) function) to explain why ln(?27) = ?_1^3 3/(2x) dx. 

          6. (a) A square pyramid crystal has a square base with sides of length a and the height of the top vertex from the base given by h. The volume of the pyramid crystal is given by V = a^2 h / 3.

[diagram]

The crystal is growing in volume at a constant rate of 0.5 cm^3 per hour. At a particular point in time the crystal has height h = 1 cm and side a = ?2 cm. This ratio of height to side is constant as the crystal grows. How fast is the height increasing at this particular point in time?

(b) Use the results from Section 3.4.1 (page 79) of your calculus notes (together with the properties of the ln(x) function) to explain why ln(?27) = ?_1^3 3/(2x) dx.
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6. (a) A square pyramid crystal has a square base with sides of length a and the height of the top vertex from the base given by h. The volume of the pyramid crystal is given by V = a^2 h / 3. [diagram] The crystal is growing in volume at a constant rate of 0.5 cm^3 per hour. At a particular point in time the crystal has height h = 1 cm and side a = ?2 cm. This ratio of height to side is constant as the crystal grows. How fast is the height increasing at this particular point in time? (b) Use the results from Section 3.4.1 (page 79) of your calculus notes (together with the properties of the ln(x) function) to explain why ln(?27) = ?1^3 3/(2x) dx.

Added by Carlos S.

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Calculus: Early Transcendentals
Calculus: Early Transcendentals
James Stewart 8th Edition
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Transcript

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00:01 All right, two problems here, kind of like this one.
00:02 So it gives us a whole bunch of information about this pyramid.
00:06 It's growing, but it always stays a pyramid.
00:09 The key is knowing that the length of the side over h is going to always equal square to two over one.
00:20 That's an important piece that we need because it means a is equal to the square to two times h.
00:32 Okay, so we're going to need this.
00:34 It's an important piece.
00:38 Okay.
00:39 We got our volume formula for a pyramid.
00:42 This is important.
00:45 And what we want to find, it says, how fast is the height increasing? we want to find dh over dt.
00:53 This is so we want to find d .h.
00:58 Over dt.
00:58 Well, what we want to do is first use this guy to put everything in terms of h.
01:04 So our volume is going to be squared of 2 times h squared times h over 3.
01:13 And now here's what i'm going to start doing some math.
01:16 So my volume is 2h cubed over 3.
01:21 Okay.
01:22 And this is nice because now i can put, i can put the, i take the derivative.
01:32 So take the derivative with respect to t, d, v, over.
01:35 D t is equal to 6h squared over 3 times d h over d t and remember d h over d t is what we're looking for we can clean this up dv over d t equals 2 h squared times d h over d t and now we just got to put our numbers in okay it's saying the crystal is growing at a constant rate of 0 .5 centimeters cube that's our dv d t that's this guy.
02:08 0 .5 centimeters cubed growing, so it's positive.
02:13 It says at a particular point, the height is one.
02:18 Inside the ratio, it says how fast is the height increasing at this particular point? so when the height is one, i can go 2 times 1 squared times dh over dt.
02:31 Well, two times one squared.
02:33 This right here is just two.
02:34 So i just have to take the centimeter squared.
02:44 I like to put the units.
02:46 So i'm squaring.
02:49 Remember this is one centimeters.
02:55 And we're squaring this thing.
02:57 So the centimeters get squared.
03:00 It's just two centimeters squared...
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