Help me on Calc 3?

Any help is great, thanks!

6. A light is at the top of a pole 80ft high. A ball is dropped is dropped from the same height 20 feet away from the light. Find how fast the shadow of the ball is moving along the ground. a) 1 second later b) 2 seconds later. (Assume the ball falls s=16t^2 ft in t seconds)

18. Consider the function y= x^m(1-x)^n, where m and n are positive integers. Show that:

a) if m is even, y has a min at x=0

b) if n is even, y has a min at x=1

c) y has a max at x=m/(m+n) regardless of m and n are even or odd.

30. A lab scientist performs an experiment n times to measure a physical quantity x and obtains the n results, x1, x2, ..., xn. These measurements deviate from the true value of x because of environmental factors. She estimates t for x based on the method of least squares. this means t is chosen to minimize the quantity

s= (t-x1)^2 + (t-x2)^2 +...+ (t-xn)^2

Show that this estimate t is the average of the measured values.

t= (x1 + x2 + ... +xn) /n

Thanks again!

Help me on Calc 3?symphony

length of the shadow=rt(80^2-h^2)

h=1/2*32*t^2

length of the shadow=rt(80^2-16t^2)

dl/dt=-32t/2(80^2-16t^2)^1/2

a)1sec later plugging in

=-32/2(6400-16)^1/2

=-32/2*80 '/sec=1/5'/sec

b)2 seconds later

=64/2(6400-64)^1/2=64/160=2/5'/sec

Help me on Calc 3?performing arts center opera theater

18. let y=x^m(1-x)^n

dy/dx=mx^(m-1)(1-x)^n-n(1-x)^(n-1)x^m

=x^(m-1)(1-x)^(n-1)[m(1-x)-nx]

=x^(m-1)(1-x)^(n-1)[m-(m+n)x]

dy/dx=0 gives x=0 or x=1 or x=m/(m+n)

(a) for x=0, consider x=-0.1 and x=0.1

if m is even. when x= -0.1, (-0.1)^(m-1) is %26lt;0 since m-1 is odd.

so dy/dx is %26lt;0 when x=-0.1

when x=0.1, dy/dx%26gt;0

therefore, x=0 is a min point.

(b) for x=1, consider x=0.9 and x=1.1

if n is even, when x=0.9, dy/dx%26lt;0 since

m-0.9(m+n)= 0.1m-0.9n%26lt;0

when x=1.1, dy/dx%26gt;0 since

(-0.1)^(n-1) is %26lt;0 and -0.1m-1.1n%26lt;0

(c) x=m/(m+n) is a max because dy/dx%26gt;0 if you consider

x slight less than m/(m+n) and dy/dx%26lt;0 if x is slightly greater

than m/(m+n)

30. ds/dt=2(t-x1)+2(t-x2)+....+2(t-xn)

ds/dt=0 gives nt-(x1+x2+...+xn)=0

t=(x1+x2+...+xn)/n