Calculus I radius and volume of a sphere / differentials?

Question

The radius of a sphere is measured to be 3.0 inches. If the measurement is correct to within 0.01 inch, use differentials to estimate the propagated error in the volume of the sphere.

I have 4 possible answers, how do I work this problem?
a. +/- 0.000001 inch^3
b. +/- 0.36π inch^3
c. +/- 0.036π inch^3
d. +/- 0.06 inch^3

Answer 1

OK, let p be pi, OK?

The volume of the sphere is 4*p*r^3/3...

NOW, the first derivative of that, with respect to the radius, is:

4*p*r^2 dr

SO, you can approximate the answer by:

(4*p*3^2)*0.01

That'd give you 0.36*pi cubic inches, which would be b

Answer 2

Assuming the pyramid would opt to opt to be a accepted sq. pyramid, the biggest sq. pyramid interior a dice with element length d, would have maximum recommendations-blowing f = d and base element e = d (such that it stocks a dice's element as its base), and its apex on the middle of the distinct dice element, such that its quantity V[pyramid] = (a million/3) e^2 f = d^3 / 3 = one-0.33 the quantity V[dice] of the dice. for that reason, V[pyramid] is maximized as V[dice] is maximized. A dice interior a cylinder with radius b and proper c, would have quantity V[dice] such that: V[dice] = min(2b, c)^3 *** Eq. a million because c is inversely proportional Assuming both caps of the cylinder are such that their round perimeters are parallel "small circles" of the section, (it particularly is, for the cylinders optimal radius b[max] = a million, c = 0; and also, for the cylinder's optimal maximum recommendations-blowing c[max] = 2, b = 0). enable ingredient o be the middle of the section, and enable ingredient p be on the fringe of a round cap. enable oq be a radius of the cylinder such that pq is perpendicular to the round cap. Then, op = the round radius a = a million, pq = a million/2 the cylinder's maximum recommendations-blowing = c/2, and oq = the cylinder's radius b opq is a suitable triangle with hypotenuse op. via potential of the Pythagorean Theorem, c^2 / 4 + b^2 = a million => c = 2 sqrt(a million - b^2) *** Eq. 2 because 0 < b < a million and 0 < c < 2, we make sure only evaluate the positive branches of the sqrts. Now, b is inversely proportional to c, and both are monotonic, that signifies that (via potential of Eq. a million), V[dice] is maximized the placement b = c. So our Pythagorean equation would opt to maximum recommendations-blowing be simplified to: b^2 / 4 + b^2 = a million => b = 2 sqrt(5) / 5 And, so V[pyramid, max] = V[dice, max] / 3 = b^3 / 3 = (2 sqrt(5) / 5)^3 / 3 = 8 sqrt(5) / seventy 5 *** answer ? 5.7% of the quantity of the section --- through creating use of exact reality the inradius of a unit dice is a million/2 and a circumradius is sqrt(3)/2, i'd opt to imagine of packing the section contained contained contained in the dice would opt to maximise section utilization. the important awkward quantity to fill seems the pyramid. So i'd opt to guess a maximal order would opt to bypass like this (from innermost to outermost): pyramid, cylinder, dice, sphere i'd opt to guess that a minimum order will be cylinder, dice, sphere, pyramid

Answer 3

Wow, that was a while ago for me. Is this homework?

You know that the volume of a sphere is given by

v = 4/3 π r*3

Right? If I recall correctly, what you need to do is differentiate with respect to "r", and evaluate for r=3 and r=3.01.

Hope that gets you started, good luck!