Find the equation of the normal to the curve at the given value of x.

answer:

explanation

the normal is perpendicular to the tangent. we need to find the slope of the tangent line to at and then take the negative reciprocal of that, which will give us a perpendicular slope to the tangent (the normal slope).

at x=-1, the y-coordinate is

so the point at which this normal occurs is (-1,-8).

to find tangent slope, we differentiate the curve with respect to x. we can use the power rule for differentiation, which says that :

the slope of the tangent at x = -1 is

so the normal slope is the negative reciprocal of that: 1/4.

with point-slope form of a line, , and knowing that this normal is at (-1,-8), the equation is

side note:

we can get an equivalent equation. if we distribute

and multiply both sides by 4 to get rid of denominator

and then isolate one side, we get

(a) the volume of the cone is 20/3 in3

hence, option a is correct.