Question

Calculus help! I know the answers but idk how to get them, pic below. I just need b,c,d

Answer 1

Second to last line, does that say \(h''(t)>0\) ?

Answer 2

yes

Answer 3

Using a linear approximation, you can show that
\[h(t)\approx h'(a)(t-a)+h(a)\]
for some known \(a\) close to a desired \(t\). In this case, you use \(a=5\) and \(t=4\). The last sentence tells you that \(h(a)=h(5)=6\), and the table says that \(h'(5)=0.7\). So,
\[h(4)\approx 0.7(4-5)+6\]
I'm not so sure about the "true value" part, though.

Answer 4

for part a it says the answer is 5.3 inches, less than true value

Answer 5

thank you:) any idea how to do parts b,c,d?

Answer 6

you have a macbook???

Answer 7

yes

Answer 8

I got to thinking about this problem a while ago, and the way to figure out the "true value" is by looking at the given values of \(h'(t)\). Since \(h''(t)>0\), you know that \(h'(t)\) must be increasing over the entire interval (as it's shown in the table). Judging by the values, you have \(h'(t)>0\) for the interval, which in turn means \(h(t)\) is increasing.

So, since 4 comes before 5, the true value is indeed smaller than the estimated value.

Answer 9

yea i just need parts b,c,d

Answer 10

Right, sorry about that, didn't notice that in the description.

The volume is given by \(V=lwh\), where \(h\) denotes the depth of water given by \(h(t)\), and the length \(l\) and width \(w\) of the container is a constant 10 and 20 (either respectively or vice versa).

\(h\) is a function of time, \(h(t)\), so \(V=20\cdot10\cdot h(t)\). So, the rate of change of volume is given by
\[\frac{dV}{dt}=200h'(t)\]
(B) asks for \(\dfrac{dV}{dt}\) when \(t=2\), which is thus
\[\frac{dV}{dt}\bigg|_{t=2}=200\cdot0.5=100\]

Answer 11

yay thats what it says in my answer booklet

Answer 12

Oh good, does the reasoning make sense?

Answer 13

yes

Answer 14

for part b how did you get 0.5 ?

Answer 15

To find \(\dfrac{dV}{dt}\) when \(t=2\), you have from the equation above,
\[\frac{dV}{dt}=20\cdot10\cdot h'(2)\]
\(h'(2)\) is given in the table.

Answer 16

ooh yeaa! lol my bad

Answer 17

any idea how to do c and d ?

Answer 18

Suppose this is a graph of \(h'(t)\) with the five desired "left-hand" rectangles drawn: