# Thread: Mean Value theorem frustration

1. ## Mean Value theorem frustration

My question is: Prove that 1/9 <= sqrt(66)-8 <= 1/8, without solving for sqrt(66). The only hint I have is to use the mean value theorem. Please don't cheat, as I have to show my work, and this question is due in 24 hours.

2. ## Re: Mean Value theorem frustration

I'm not sure how to do this using the 'mean value theorem', I'm pretty sure the mean value theorem goes something like: (roughly) when considering an arc between two endpoints of a function, at least one point exists where tangent to the arc is parallel to the line connecting the endpoints.

However, this is how I would approach your problem:

$\tiny \dpi{200} \fn_cm \frac{1}{9}\leq \sqrt{66}-8\leq \frac{1}{8}$

$\tiny \dpi{200} \fn_cm \frac{1}{9}(\sqrt{66}+8)\leq (\sqrt{66}-8)(\sqrt{66}+8)\leq \frac{1}{8}(\sqrt{66}+8)$

$\tiny \dpi{200} \fn_cm \frac{1}{9}\sqrt{66}+\frac{8}{9}\leq 2\leq \frac{1}{8}\sqrt{66}+1$

With the expression in this form it's easier to argue the following:

$\tiny \dpi{200} \fn_cm \frac{1}{9}\sqrt{66}+\frac{8}{9}< \frac{1}{9}\sqrt{81}+\frac{8}{9}$

$\tiny \dpi{200} \fn_cm \frac{1}{9}\sqrt{66}+\frac{8}{9}< \frac{17}{9}$

As well as:

$\tiny \dpi{200} \fn_cm \frac{1}{8}\sqrt{64}+1< \frac{1}{8}\sqrt{66}+1$

$\tiny \dpi{200} \fn_cm 2< \frac{1}{8}\sqrt{66}+1$

So:

$\tiny \dpi{200} \fn_cm \because \frac{1}{9}\sqrt{66}+\frac{8}{9}< \frac{17}{9}< 2\leq 2< \frac{1}{8}\sqrt{66}+1$

It must be true that:

$\tiny \dpi{200} \fn_cm \frac{1}{9}\sqrt{66}+\frac{8}{9}\leq 2\leq \frac{1}{8}\sqrt{66}+1$

$\tiny \dpi{200} \fn_cm \Rightarrow \frac{1}{9}\leq \sqrt{66}-8\leq \frac{1}{8}$

3. ## Re: Mean Value theorem frustration

Thanks a lot! I imagine that you're still on the site, so here's another:

The area between two varying concentric circles is at all times 9pi inches^2. The rate of change of the area of the larger circle is 10pi inches^2/sec. How fast is the circumference of the smaller circle changing when it has area 16pi inches^2?

4. ## Re: Mean Value theorem frustration

Im using large letters for larger circle and small letters for smaller circle
A-a=9pi dA/dt-da/dt=0 dA/dt=10 So da/dt=10 Whena=16pi pi(r^2)=16pi So r=4
We want dc/dt
dc/dt=da/dt*dc/da c=2pir and a=pir^2 so get dc/da from dc/dr*dr/da and finish off