Mathematics, 10.03.2020 22:32 juanmercs99

A rectangular solid with a square base has a volume of 2,744 cubic inches. (Let w represent the length of the sides of the square base and let h represent the height of the solid.) (a) Determine the dimensions (in inches) that yield the minimum surface area.

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Answer from: hey840

For minimum surface area, the base length and height are equal in length and equal to 14 inches.

Step-by-step explanation:

Given:

Volume of rectangular solid (V) = 2744 cubic inches

Length of base side = 'w'

Height of solid = 'h'

We know that,

Volume = Area of base × Height

$2744=(w\times w)\times h\\\\2744=w^2h\\\\h=\frac{2744}{w^2}-----(1)$

Now, surface area of the solid is given as:

$Surface\ area=2\times (base\ area)+4(hw)\\\\Surface\ area=2w^2+4hw$

Plug in 'h' value from equation (1). This gives,

$Surface\ area=2w^2+4\times\frac{2744w}{w^2}\\\\Surface\ area=2(w^2+\frac{5488}{w})$

Now, for minimum surface area, the derivative of surface area with respect to length 'w' must be 0.

Differentiating both sides with respect to 'w' and equating it to 0, we get:

$\frac{dS}{dw}=0\\\\\frac{d}{dw}[2(w^2+\frac{5488}{w})]=0\\\\2w-\frac{5488}{w^2}=0\\\\2w=\frac{5488}{w^2}\\\\w^3=\frac{5488}{2}\\\\w=\sqrt[3]{2744}= 14\ in$

Therefore, the base length is 14 inches.

Now, from equation (1), the height is given as:

$h=\frac{2744}{w^2}\\\\h=\frac{2744}{14^2}=14\ in$

Therefore, for minimum surface area, the base length and height are equal in length and equal to 14 inches.

Answer from: Quest

the degrees would remain the same ( the number to the left of the decimal point.

the multiply the decimal by 60 to get the minutes and use the numbers to the left of the decimal point.

then multiply the decimal portion of that by 60 for the seconds and round off as needed.

39 .956652

the 39 remains the degrees.

the minutes = 0.956652 x 60 = 57.39912

the minutes = 57

the seconds = 0.39912 x 60 = 23.95 = 24 ( rounded to nearest tenth)

so 39.956652 = 39 degrees, 57 minutes and 24 seconds

-75.159809

the -75 remains the degrees

the minutes = 0.159809 x 60 = 9.58854

the minutes = 9

the seconds = 0.58854 x 60 = 35.31, rounded to nearest tenth = 35.3 seconds

so -75.159809 = -75 degrees, 9 minutes and 35.3 seconds.

the first coordinate is north or south, since it it positive it is north.

the second number is east or west, because it is negative it is west.

39 degrees, 57 minutes and 24 seconds n

-75 degrees, 9 minutes and 35.3 seconds w

Answer from: Quest

answer:

given the linear inequality : $x-2y\geq -12$

add 2y to both sides we get;

$x-2y+2y\geq -12+2y$

simplify:

$x\geq -12+2y$

add 12 to both sides of an equation;

$x+12\geq -12+2y+12$

simplify:

$x+12\geq 2y$

divide both sides by 2 we get;

$\frac{x}{2} + 6 \geq y$

$y\leq \frac{x}{2} + 6$

the related equation for the inequality equation [1] is;

$y =\frac{x}{2} +6$

y-intercepts defined as the graph crosses the y-axis.

substitute x =0 in [2] to solve for y;

$y =\frac{0}{2} +6$

y = 6

x-intercepts defined as the graph crosses the x-axis.

substitute value y = 0 to solve for x;

$0 =\frac{x}{2} +6$

subtract 6 from both sides we get

$0-6 =\frac{x}{2} +6-6$

$-6 =\frac{x}{2}$

multiply both sides by 2 we get;

$-6 \times 2=x$

x = -12

now, plot these point (0, 6) and (-12, 0) on graph as shown below and shade the graph below the solid line.

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A rectangular solid with a square base has a volume of 2,744 cubic inches. (Let w represent the leng...