$Use:\\\\(ab)^n=a^nb^n\\\\(a^n)^m=a^{nm}\\\\a^n\cdot a^m=a^{n+m}\\\\\sqrt[n]{a}=a^\frac{1}{n}\\------------------------\\\\\left(16n^4\right)^{\frac{5}{4}}=16^\frac{5}{4}\left(n^4\right)^\frac{5}{4}=16^{1\frac{1}{4}}n^{4\cdot\frac{5}{4}}=16^{1+\frac{1}{4}}n^5=16^1\cdot16^\frac{1}{4}\cdot n^5\\\\=16\cdot\sqrt[4]{16}\cdot n^5=16\cdot2\cdot n^5=\boxed{32n^5}\\\\\sqrt[4]{16}=2\ because\ 2^4=16\\\\\text{Answer}\ \boxed{\left(16n^4\right)^\frac{5}{4}=32n^5}$

4 0

3 years ago

A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 70 cm/s. Find the rate at which th

jeyben [28]

Answer:

(A) 30772cm^2/s

(B) 153860cm^2/s

Step-by-step explanation:

We are given

speed of the ripple = 70cm/s

this speed is increasing the radius which means speed = radius /time

radius = speed*time

r= 70t

Now we know the area of the circle

A = π $r^2$

this area will be increasing as ripples in water will spread out

upon differentiating

dA/dt = 2πr * $\frac{dr}{dt}$

dA/dt = 2π*70t*70 ( r= 70t , dr/dt = speed = 70)

dA/dt = 9800π*t

(A) after 1 second

dA/dt = 9800π*1=9800π = 30772cm^2/s

(B) after 5 seconds

dA/dt = 9800π*5 = 153860cm^2/s

(C) after 6 seconds

dA/dt = 9800π*6 = 184632 cm^2/s

4 0

3 years ago