Lesson Minimize surface area of a conical paper cup with the given volume
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<H2>Minimize surface area of a conical paper cup with the given volume</H2> <H3>Problem 1</H3>A cone-shaped paper drinking cup is to hold 40 cm^3 of water. Find the height and radius of the cup that will require the least amount of paper. <B>Solution</B> <pre> The volume of the cone cup is given 40 cm^3. The lateral area of a cone is S = {{{pi*r*s}}}, (1) where r is the base radius and "s" is the slant height: s = {{{sqrt(h^2 + r^2)}}}. So, we need minimize the lateral area S = {{{pi*r*sqrt(h^2+r^2)}}} (2) at given restriction for the volume {{{(1/3)*pi*r^2*h}}} = 40 cm^3. (3) From the restriction (3), h = {{{120/(pi*r^2)}}}. We substitute it into expression (2), and we get S(r) as a function of the radius r, only S(r) = {{{pi*r*sqrt((14400/(pi^2*r^4)) + r^2)}}} = {{{sqrt((14400/r^2) + pi^2*r^4)}}} +-------------------------------------------------------+ | To find the minimum of S(r), we should calculate | | the derivative and equate it to zero. | +-------------------------------------------------------+ I will not calculate the derivative in full, which is a complicated fraction. It is enough to calculate its numerator and equate it to zero. It gives this equation -{{{2*(14400/r^3)}}} + {{{4*pi^2r^3}}} = 0, or, equivalently 7200 = {{{pi^2*r^6}}}, {{{r^6}}} = {{{7200/pi^2}}}, r = {{{root(6,7200/pi^2)}}} = 3.0 cm. (4) So, the radius is just found. The height should be h = {{{120/(pi*r^2)}}} = {{{120/(pi*3^2)}}} = 4.24 cm. (5) Expressions (4) and (5) give the required exact formulas and approximate numerical values. </pre> My other lessons on Calculus word problems at this site are - <A HREF=https://www.algebra.com/algebra/homework/word/misc/A-ladder-foot-slides-on-the-ground-txt.lesson>A ladder foot slides on the ground</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Finding-rate-of-change-of-some-processes.lesson>Finding rate of change of some processes</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Find-the-derivative-of-a-function-defined-by-complicated-expression.lesson>Find the derivative of a function defined by complicated expression</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Taking-derivative-of-a-function-which-is-defined-implicitly.lesson>Taking derivative of a function, which is defined implicitly</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Find-the-derivative-of-a-function-satisfying-given-functional-equation.lesson>Find the derivative for a function satisfying given functional equation</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/09-add-Find-the-Range-of-f%28x%29-=-%285cos%28x%29%29-div-%28x-%2B-1%29%29cos%28x%29.lesson>Find the range of f(x) = (5*cos(x))/(x + 1)), x >=0</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/A-tricky-Calculus-problem-on-derivative-and-anti-derivative.lesson>A tricky Calculus problem on derivative and anti-derivative</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Couple-of-non-standard-Calculus-problems.lesson#google_vignette>Couple of non-standard Calculus problems</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Finding-the-minimum-of-a-function-defined-on-a-curve-in-the-coordinate-plane.lesson>Finding the minimum of a function defined on a curve in a coordinate plane</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Tricky-solution-to-find-the-maximum-of-a-function-defined-by-a-complicated-expression.lesson>Tricky solution to find the maximum of a function defined by a complicated expression</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Maximize-the-area-of-a-trapezoid.lesson>Maximize the area of a trapezoid </A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Maximize-the-area-of-a-trapezoid.lesson>Maximize the area of a trapezoid </A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Maximmize-the-volume-of-an-open-box.lesson>Maximize the volume of an open box</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Minimize-surface-area-of-a-rectangular-prizm-with-the-given-volume.lesson>Minimize surface area of a rectangular box with the given volume</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Minimize-the-cost-of-an-aquarium-with-the-given-volume.lesson>Minimize the cost of an aquarium with the given volume</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Find-the-volume-of-a-solid-body-obtained-by-rotation-the-area-about-an-axis.lesson>Find the volume of a solid obtained by rotation of some plane shape about an axis</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Finding-the-volume-of-a-solid-body-mentally.lesson>Finding the volume of a solid body mentally</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/OVERVIEW-of-my-lessons-on-Calculus-word-problems.lesson>OVERVIEW of my lessons on Calculus word problems</A> Use this file/link <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A> to navigate over all topics and lessons of the online textbook ALGEBRA-II.
