aanpalendmw

2022-07-18

Finding dimensions of a rectangular box to optimize volume

A rectangular box (base is not square) with an open top must have a length of 3ft, and a surface area of $16f{t}^{2}$. Compute the dimensions of the box that will maximize its volume.

I am going wrong somewhere but I can't see where. I have set $2(3h+3w+hw)=16$ and my optimization equation is $v=hwl$. I set $w=\frac{8-3h}{3+h}$ and plug into the optimization equation. Calculating v' gives me $\frac{9{h}^{2}-84h+72}{{(3+h)}^{2}}$,but setting that equal to zero does not give me the answer. I need for h, which is 1.

A rectangular box (base is not square) with an open top must have a length of 3ft, and a surface area of $16f{t}^{2}$. Compute the dimensions of the box that will maximize its volume.

I am going wrong somewhere but I can't see where. I have set $2(3h+3w+hw)=16$ and my optimization equation is $v=hwl$. I set $w=\frac{8-3h}{3+h}$ and plug into the optimization equation. Calculating v' gives me $\frac{9{h}^{2}-84h+72}{{(3+h)}^{2}}$,but setting that equal to zero does not give me the answer. I need for h, which is 1.

Abbigail Vaughn

Beginner2022-07-19Added 15 answers

Explanation:

The equation is going wrong you forgot to subtract the top so itll be $2(3b+bh+3h)-3b=16$ here b=breadth and h=height . Now you proceed by the way as you did and you will get an equation in h for volume diffrentiate it and put it to be 0. Thus we get eqn $\frac{48h-18{h}^{2}}{3+2h}$ so on differentiating wrt h we get a quadratic as ${h}^{2}+2h-4=0$ thus $h=\frac{-1+,-\sqrt{4+16}}{2}$ so we take positive sign for max area thus its $-1+\sqrt{5}$ now you can get breadth and you are done with the sum.

The equation is going wrong you forgot to subtract the top so itll be $2(3b+bh+3h)-3b=16$ here b=breadth and h=height . Now you proceed by the way as you did and you will get an equation in h for volume diffrentiate it and put it to be 0. Thus we get eqn $\frac{48h-18{h}^{2}}{3+2h}$ so on differentiating wrt h we get a quadratic as ${h}^{2}+2h-4=0$ thus $h=\frac{-1+,-\sqrt{4+16}}{2}$ so we take positive sign for max area thus its $-1+\sqrt{5}$ now you can get breadth and you are done with the sum.

comAttitRize8

Beginner2022-07-20Added 2 answers

Step 1

Did you forget that the top of the box is open? The surface area is $2(3h+hw)+3w=16,$ so that $w=\frac{16-6h}{3+2h}.$.

Step 2

The volume of the box is $V=3hw=3h\frac{16-6h}{3+2h}.$

If you differentiate this with respect to h, and set it equal to 0, you get the h for which the volume is maximized.

Did you forget that the top of the box is open? The surface area is $2(3h+hw)+3w=16,$ so that $w=\frac{16-6h}{3+2h}.$.

Step 2

The volume of the box is $V=3hw=3h\frac{16-6h}{3+2h}.$

If you differentiate this with respect to h, and set it equal to 0, you get the h for which the volume is maximized.

The distance between the centers of two circles C1 and C2 is equal to 10 cm. The circles have equal radii of 10 cm.

A part of circumference of a circle is called

A. Radius

B. Segment

C. Arc

D. SectorThe perimeter of a basketball court is 108 meters and the length is 6 meters longer than twice the width. What are the length and width?

What are the coordinates of the center and the length of the radius of the circle represented by the equation ${x}^{2}+{y}^{2}-4x+8y+11=0$?

Which of the following pairs of angles are supplementary?

128,62

113,47

154,36

108,72What is the surface area to volume ratio of a sphere?

An angle which measures 89 degrees is a/an _____.

right angle

acute angle

obtuse angle

straight angleHerman drew a 4 sided figure which had only one pair of parallel sides. What could this figure be?

Trapezium

Parallelogram

Square

RectangleWhich quadrilateral has: All sides equal, and opposite angles equal?

Trapezium

Rhombus

Kite

RectangleKaren says every equilateral triangle is acute. Is this true?

Find the number of lines of symmetry of a circle.

A. 0

B. 4

C. 2

D. InfiniteThe endpoints of a diameter of a circle are located at (5,9) and (11, 17). What is the equation of the circle?

What is the number of lines of symmetry in a scalene triangle?

A. 0

B. 1

C. 2

D. 3How many diagonals does a rectangle has?

A quadrilateral whose diagonals are unequal, perpendicular and bisect each other is called a.

A. rhombus

B. trapezium

C. parallelogram