Its simply 87 + 30, which equals 117.
Each year, it's value is 1.08 of the previous price, so the equation is
![x=32000*(1.08^6)](https://tex.z-dn.net/?f=x%3D32000%2A%281.08%5E6%29)
Using a calculator, you get that the answer is 50779.978. You must also round to the nearest hundredth because in our currency, cents are the smallest unit.
Therefore, an antique car that costs $32000 will be worth 50779.98 6 years after.
Answer:
216 ft
Step-by-step explanation:
area of a rectangle is L(ength) * W(idth)
L=12 W=18
12 * 18 = 216
If
![f(x) = e^{ax}\cos(bx)](https://tex.z-dn.net/?f=f%28x%29%20%3D%20e%5E%7Bax%7D%5Ccos%28bx%29)
then by the product rule,
![f'(x) = \left(e^{ax}\right)' \cos(bx) + e^{ax}\left(\cos(bx)\right)'](https://tex.z-dn.net/?f=f%27%28x%29%20%3D%20%5Cleft%28e%5E%7Bax%7D%5Cright%29%27%20%5Ccos%28bx%29%20%2B%20e%5E%7Bax%7D%5Cleft%28%5Ccos%28bx%29%5Cright%29%27)
and by the chain rule,
![f'(x) = e^{ax}(ax)'\cos(bx) - e^{ax}\sin(bx)(bx)'](https://tex.z-dn.net/?f=f%27%28x%29%20%3D%20e%5E%7Bax%7D%28ax%29%27%5Ccos%28bx%29%20-%20e%5E%7Bax%7D%5Csin%28bx%29%28bx%29%27)
which leaves us with
![f'(x) = \boxed{ae^{ax}\cos(bx) - be^{ax}\sin(bx)}](https://tex.z-dn.net/?f=f%27%28x%29%20%3D%20%5Cboxed%7Bae%5E%7Bax%7D%5Ccos%28bx%29%20-%20be%5E%7Bax%7D%5Csin%28bx%29%7D)
Alternatively, if you exclusively want to use the chain rule, you can carry out logarithmic differentiation:
![\ln(f(x)) = \ln(e^{ax}\cos(bx)} = \ln(e^{ax})+\ln(\cos(bx)) = ax + \ln(\cos(bx))](https://tex.z-dn.net/?f=%5Cln%28f%28x%29%29%20%3D%20%5Cln%28e%5E%7Bax%7D%5Ccos%28bx%29%7D%20%3D%20%5Cln%28e%5E%7Bax%7D%29%2B%5Cln%28%5Ccos%28bx%29%29%20%3D%20ax%20%2B%20%5Cln%28%5Ccos%28bx%29%29)
By the chain rule, differentiating both sides with respect to <em>x</em> gives
![\dfrac{f'(x)}{f(x)} = a + \dfrac{(\cos(bx))'}{\cos(bx)} \\\\ \dfrac{f'(x)}{f(x)} = a - \dfrac{\sin(bx)(bx)'}{\cos(bx)} \\\\ \dfrac{f'(x)}{f(x)} = a-b\tan(bx)](https://tex.z-dn.net/?f=%5Cdfrac%7Bf%27%28x%29%7D%7Bf%28x%29%7D%20%3D%20a%20%2B%20%5Cdfrac%7B%28%5Ccos%28bx%29%29%27%7D%7B%5Ccos%28bx%29%7D%20%5C%5C%5C%5C%20%5Cdfrac%7Bf%27%28x%29%7D%7Bf%28x%29%7D%20%3D%20a%20-%20%5Cdfrac%7B%5Csin%28bx%29%28bx%29%27%7D%7B%5Ccos%28bx%29%7D%20%5C%5C%5C%5C%20%5Cdfrac%7Bf%27%28x%29%7D%7Bf%28x%29%7D%20%3D%20a-b%5Ctan%28bx%29)
Solve for <em>f'(x)</em> yields
![f'(x) = e^{ax}\cos(bx) \left(a-b\tan(bx)\right) \\\\ f'(x) = e^{ax}\left(a\cos(bx)-b\sin(bx))](https://tex.z-dn.net/?f=f%27%28x%29%20%3D%20e%5E%7Bax%7D%5Ccos%28bx%29%20%5Cleft%28a-b%5Ctan%28bx%29%5Cright%29%20%5C%5C%5C%5C%20f%27%28x%29%20%3D%20e%5E%7Bax%7D%5Cleft%28a%5Ccos%28bx%29-b%5Csin%28bx%29%29)
just as before.