The radius of the puddle is zero when time is zero but expands at a rate of one-quarter unit per second instead of one-half unit per second. By what factor does the denominator of the resultant composite function A(r(t)) change from that in the opening scenario?

Given that the radius of the puddle expands, originally, at a rate of 1/4 unit/second instead of 1/2 unit/second.

The current rate of expension of the radius,

\frac {r(t)}{t}=frac 1 4 unit/second ...(i)

The resultant composite function is area, A(r(t)).

The area of a circular region having radius r is,

If the radius is the function of time, then the composite function is

To determine the rate of change of area with respect to time, differentiate the composite function A(r(t)) once with respect to time, we have

So, the composite function is changing twice of the rate of change of the radius.

By using equation (i), if r(i) is changing with R= 1/4 unit / second.

Then, A(r(t)) will change 2 x 1/4 = 1/2 square unit/second.

x= -3 and

step-by-step explanation:

to find the roots of a cubic, use division or factoring to find the factors then set equal to 0. as this is a difference of cubes, it factors into

where a is the cube root of .

here   this is 27 and the cube root of 27 is 3. so we write using the form:

.

set each factor to 0 and solve for x.

x-3=0

x=3

the quadratic expression does not factor and must be solved using the quadratic formula.

where a = 1, b=3, and c=9

this cubic has two complex roots.