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3 years ago

A circle has a radius of 6 inches and a central angle of 60°. What is the measure of the arc length associated with this angle?

Mathematics

1 answer:

Readme [11.4K]3 years ago

6 0

Answer:

$2\pi$

Step-by-step explanation:

first you take 60°/360°= 1/6 then you take 1/6 and multiple it by $2\pi$ and that equals $\frac{\pi }{3}$ finally take $\frac{\pi }{3}$ times the radius (6) and that should equal $2\pi$

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B) Let g(x) =x/2sqrt(36-x^2)+18sin^-1(x/6)<br><br> Find g'(x) =

jolli1 [7]

I suppose you mean

$g(x) = \dfrac x{2\sqrt{36-x^2}} + 18\sin^{-1}\left(\dfrac x6\right)$

Differentiate one term at a time.

Rewrite the first term as

$\dfrac x{2\sqrt{36-x^2}} = \dfrac12 x(36-x^2)^{-1/2}$

Then the product rule says

$\left(\dfrac12 x(36-x^2)^{-1/2}\right)' = \dfrac12 x' (36-x^2)^{-1/2} + \dfrac12 x \left((36-x^2)^{-1/2}\right)'$

Then with the power and chain rules,

$\left(\dfrac12 x(36-x^2)^{-1/2}\right)' = \dfrac12 (36-x^2)^{-1/2} + \dfrac12\left(-\dfrac12\right) x (36-x^2)^{-3/2}(36-x^2)' \\\\ \left(\dfrac12 x(36-x^2)^{-1/2}\right)' = \dfrac12 (36-x^2)^{-1/2} - \dfrac14 x (36-x^2)^{-3/2} (-2x) \\\\ \left(\dfrac12 x(36-x^2)^{-1/2}\right)' = \dfrac12 (36-x^2)^{-1/2} + \dfrac12 x^2 (36-x^2)^{-3/2}$

Simplify this a bit by factoring out $\frac12 (36-x^2)^{-3/2}$ :

$\left(\dfrac12 x(36-x^2)^{-1/2}\right)' = \dfrac12 (36-x^2)^{-3/2} \left((36-x^2) + x^2\right) = 18 (36-x^2)^{-3/2}$

For the second term, recall that

$\left(\sin^{-1}(x)\right)' = \dfrac1{\sqrt{1-x^2}}$

Then by the chain rule,

$\left(18\sin^{-1}\left(\dfrac x6\right)\right)' = 18 \left(\sin^{-1}\left(\dfrac x6\right)\right)' \\\\ \left(18\sin^{-1}\left(\dfrac x6\right)\right)' = \dfrac{18\left(\frac x6\right)'}{\sqrt{1 - \left(\frac x6\right)^2}} \\\\ \left(18\sin^{-1}\left(\dfrac x6\right)\right)' = \dfrac{18\left(\frac16\right)}{\sqrt{1 - \frac{x^2}{36}}} \\\\ \left(18\sin^{-1}\left(\dfrac x6\right)\right)' = \dfrac{3}{\frac16\sqrt{36 - x^2}} \\\\ \left(18\sin^{-1}\left(\dfrac x6\right)\right)' = \dfrac{18}{\sqrt{36 - x^2}} = 18 (36-x^2)^{-1/2}$

So we have

$g'(x) = 18 (36-x^2)^{-3/2} + 18 (36-x^2)^{-1/2}$

and we can simplify this by factoring out $18(36-x^2)^{-3/2}$ to end up with

$g'(x) = 18(36-x^2)^{-3/2} \left(1 + (36-x^2)\right) = \boxed{18 (36 - x^2)^{-3/2} (37-x^2)}$

5 0

2 years ago

A magazine currently has 8700 subscribers for its online web version. It is adding members at the rate of R(t) = 190e0.03t subsc

aleksklad [387]

Number of subscriber the magazine will have after 3 years from now approximately be 8767

<u>Solution:</u>

Given that magazine currently has 8700 subscribers for its online web version

$\begin{array}{l}{\mathrm{R}(\mathrm{t})=190 \mathrm{e}^{0.03 \mathrm{t}} \text { subscribers/month }} \\\\ {\mathrm{S}(\mathrm{t})=\mathrm{e}^{-0.06 \mathrm{t}}}\end{array}$

After 3 years, time(t) = 36 month

Total number of subscribers after 3 years from now :

Substitute "t" = 36

$\begin{array}{l}{\mathrm{R}(36)=190 \mathrm{e}^{0.03 \times(36)}=190 \times(2.944)} \\\\ {\mathrm{R}(36) \approx 560} \\\\ {\mathrm{S}(36)=\mathrm{e}^{-0.06 \times(36)}=0.12}\end{array}$

Subscribers remaining = 0.12 x 560 = 67.2

The magazine currently has 8700 subscribers

Added Subscriber = 8700 + 560 = 9260

Remaining Subscriber = 8700 + 67.2 = 8767.2

Therefore number of subscriber the magazine will have after 3 years from now approximately be 8767

7 0

3 years ago

Please help i dont remember how to do this

inessss [21]

Just add 5 5 and 10 10 to get yo answers
i will work

8 0

2 years ago

Students are selling raffle tickets for a school fundraiser. They collect $25 for every 10 raffle tickets sold. Rain equation th

Free_Kalibri [48]

Answer: m= $25/10 r

Step-by-step explanation:

Let m= money

r= raffle ticket

Then according to the statement

m= $25 for 10 tickets

so 10 tickets= $25

Or the equation goes,

m= $25/10 r

7 0

3 years ago

Simplify the following expression by combining like terms.

Aleks04 [339]

Answer:

8x^2+3x

Step-by-step explanation:

What I read is 2x+8x^2-4x+5x

combining like terms means putting the terms that have the same variable part together

8x^2 is the only one that doesn't have any terms like it as far as the variable part

so 8x^2+2x-4x+5x

We just need to figure out 2-4+5 which is -2+5=3

So the answer is 8x^2+3x

7 0

3 years ago