Answer:
The third side is increasing at an approximate rate of about 0.444 meters per minute.
Step-by-step explanation:
We are given a triangle with two sides having constant lengths of 13 m and 19 m. The angle between them is increasing at a rate of 2° per minute and we want to find the rate at which the third side of the triangle is increasing when the angle is 60°.
Let the angle between the two given sides be θ and let the third side be <em>c</em>.
Essentially, given dθ/dt = 2°/min and θ = 60°, we want to find dc/dt.
First, convert the degrees into radians:
![\displaystyle 2^\circ \cdot \frac{\pi \text{ rad}}{180^\circ} = \frac{\pi}{90}\text{ rad}](https://tex.z-dn.net/?f=%5Cdisplaystyle%202%5E%5Ccirc%20%5Ccdot%20%5Cfrac%7B%5Cpi%20%5Ctext%7B%20rad%7D%7D%7B180%5E%5Ccirc%7D%20%3D%20%5Cfrac%7B%5Cpi%7D%7B90%7D%5Ctext%7B%20rad%7D)
Hence, dθ/dt = π/90.
From the Law of Cosines:
![\displaystyle c^2 = a^2 + b^2 - 2ab\cos \theta](https://tex.z-dn.net/?f=%5Cdisplaystyle%20c%5E2%20%3D%20a%5E2%20%2B%20b%5E2%20-%202ab%5Ccos%20%5Ctheta)
Since <em>a</em> = 13 and <em>b</em> = 19:
![\displaystyle c^2 = (13)^2 + (19)^2 - 2(13)(19)\cos \theta](https://tex.z-dn.net/?f=%5Cdisplaystyle%20c%5E2%20%3D%20%2813%29%5E2%20%2B%20%2819%29%5E2%20-%202%2813%29%2819%29%5Ccos%20%5Ctheta)
Simplify:
![\displaystyle c^2 = 530 - 494\cos \theta](https://tex.z-dn.net/?f=%5Cdisplaystyle%20c%5E2%20%3D%20530%20-%20494%5Ccos%20%5Ctheta)
Take the derivative of both sides with respect to <em>t: </em>
<em />
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Implicitly differentiate:
![\displaystyle 2c\frac{dc}{dt} = 494\sin\theta \frac{d\theta}{dt}](https://tex.z-dn.net/?f=%5Cdisplaystyle%202c%5Cfrac%7Bdc%7D%7Bdt%7D%20%3D%20494%5Csin%5Ctheta%20%5Cfrac%7Bd%5Ctheta%7D%7Bdt%7D)
We want to find dc/dt given that dθ/dt = π/90 and when θ = 60° or π/3. First, find <em>c: </em>
<em />
<em />
<em />
Substitute:
![\displaystyle 2\left(\sqrt{283}\right) \frac{dc}{dt} = 494\sin\left(\frac{\pi}{3}\right)\left(\frac{\pi}{90}\right)](https://tex.z-dn.net/?f=%5Cdisplaystyle%202%5Cleft%28%5Csqrt%7B283%7D%5Cright%29%20%5Cfrac%7Bdc%7D%7Bdt%7D%20%3D%20494%5Csin%5Cleft%28%5Cfrac%7B%5Cpi%7D%7B3%7D%5Cright%29%5Cleft%28%5Cfrac%7B%5Cpi%7D%7B90%7D%5Cright%29)
Solve for dc/dt:
![\displaystyle \frac{dc}{dt} = \frac{494\sin \dfrac{\pi}{3} \cdot \dfrac{\pi}{90}}{2\sqrt{283}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bdc%7D%7Bdt%7D%20%3D%20%5Cfrac%7B494%5Csin%20%5Cdfrac%7B%5Cpi%7D%7B3%7D%20%5Ccdot%20%5Cdfrac%7B%5Cpi%7D%7B90%7D%7D%7B2%5Csqrt%7B283%7D%7D)
Evaluate. Hence:
![\displaystyle \begin{aligned} \frac{dc}{dt} &= \frac{494\left(\dfrac{\sqrt{3}}{2} \right)\cdot \dfrac{\pi}{90}}{2\sqrt{283}}\\ \\ &= \frac{\dfrac{247\sqrt{3}\pi}{90}}{2\sqrt{283}}\\ \\ &= \frac{247\sqrt{3}\pi}{180\sqrt{283}} \\ \\ &\approx 0.444\text{ m/min}\end{aligned}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cbegin%7Baligned%7D%20%5Cfrac%7Bdc%7D%7Bdt%7D%20%26%3D%20%5Cfrac%7B494%5Cleft%28%5Cdfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D%20%5Cright%29%5Ccdot%20%5Cdfrac%7B%5Cpi%7D%7B90%7D%7D%7B2%5Csqrt%7B283%7D%7D%5C%5C%20%5C%5C%20%26%3D%20%5Cfrac%7B%5Cdfrac%7B247%5Csqrt%7B3%7D%5Cpi%7D%7B90%7D%7D%7B2%5Csqrt%7B283%7D%7D%5C%5C%20%5C%5C%20%26%3D%20%5Cfrac%7B247%5Csqrt%7B3%7D%5Cpi%7D%7B180%5Csqrt%7B283%7D%7D%20%5C%5C%20%5C%5C%20%26%5Capprox%200.444%5Ctext%7B%20m%2Fmin%7D%5Cend%7Baligned%7D)
The third side is increasing at an approximate rate of about 0.444 meters per minute.
Answer: I recommend using a site called "Desmos" it shows you where the points are. Hopefully, I helped you.
Answer:
Step-by-step explanation:
Ummm, y = mx+b is called the slope-intercept form because you can just read them off:
slope = m
y-intercept = b
Now can you pick a choice?
$4,800
If you multiply $32,000 by 15% or .15, you will receive, or in this case be taken away $4,800. Boom.
Hello!
Here are 6 steps to Help You Out!
1. 3 divided by 50 = .06
2. Then if you like you can turn it in to a percent not sure how the answer is supposed to look for example:
<span><span>3. 3/50</span>=<span>6/100</span>=6%
</span>4. 7 divided by 50 - .14
5. so 0.06 for the first one
6. And 40 divided by 50 = .8
====>So as we can see, o<span>pen side up .06 closed side up .14 and on its side .8 or open 6% closed 14% and side 80% chance.<====
Hope this Helps! Have A WONDERFUL Day! :)</span>