Mathematics, 15.10.2019 14:20 khameleonjr10

Drum tight containers is designing an open-top, square-based, rectangular box that will have a volume of 62.5 in^3. what dimension will minimize surface area? what is the minimum surface area?

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Square means L=W
V=62.5=LWH

L=W so
V=62.5=HL^2
SA=2(L^2+2LH)

we have
V=62.5=HL^2
solve for H
divide both sides by L^2
62.5/L^2=H
sub that for H in other equation

SA=2(L^2+2L(62.5/L^2))
SA=2(L^2+125/L)
SA=2L^2+250/L
find minimum of 2L^2+250L^-1
take the derivitive
4L-250L^-2, or 2(2L^3-125)/L^2
find where it equals zero
it equals zero at L=2.5∛4
L=W
if we evaluate 2L^2+250/L at L=2.5∛4, the value is 75∛2

H=62.5/L^2
H= $\frac{25 \sqrt[3]{2} }{4}$

dimentions are
L=2.5∛4
W=2.5∛4
H= $\frac{25 \sqrt[3]{2} }{4}$
minimum surface area is 75∛2 in^2 or aprox 94.4941 in^2

Answer from: taylorbean315

The dimensions of minimize surface area are $\boxed{13{\text{ in}}}{\text{ and }}\boxed{{\text{6}}{\text{.5 in}}}$ and the minimum surface area is $\boxed{507{\text{ i}}{{\text{n}}^2}}.$

Further explanation:

Given:

The volume of the rectangular box is $62.5{\text{ i}}{{\text{n}}^3}$

Explanation:

Consider the base length of the square box as “x”.

Consider the height of the box as “y”.

The surface area of the open box can be expressed as follows,

$\boxed{{\text{Surface Area}} = 4xy + {x^2}}$

The volume of the box is $62.5{\text{ i}}{{\text{n}}^3}.$

$\begin{aligned}{\text{Volume}}&= {x^2}y\\\frac{{62.5}}{{{x^2}}} &= y\\\end{aligned}$

The surface area of the box can be expressed as,

$\begin{aligned}{\text{Surface area}} &= {x^2} + 4xy\\&= {x^2} + 4x \times \frac{{62.5}}{{{x^2}}}\\&= {x^2} + \frac{{250}}{x} \\\end{aligned}$

Differentiate the surface area with respect to “x”.

$\begin{aligned}\frac{d}{{dx}}\left( {{\text{Surface area}}} \right)&= \frac{d}{{dx}}\left( {{x^2} + \frac{{4394}}{x}} \right)\\&= 2x - \frac{{250}}{{{x^2}}}\\\end{aligned}$

Substitute 0 for $\dfrac{d}{{dx}}\left( {{\text{Surface area}}} \right)$ in above equation to obtain the value of x.

$\begin{aligned}\frac{d}{{dx}}\left( {{\text{Surface area}}} \right)&= 0\\2x - \frac{{250}}{{{x^2}}}&= 0\\2x&=\frac{{250}}{{{x^2}}}\\x \times {x^2}&= \frac{{125}}{2}\\\end{aligned}$

Further solve the above equation.

$\begin{aligned}{x^3}&= \frac{{250}}{2}\\{x^3} &= 125\\x &= \sqrt[3]{{125}}\\x &= 5\\\end{aligned}$

The side of the base is $5{\text{ in}}.$

The height of the box can be obtained as follows,

$\begin{aligned}y&= \frac{{62.5}}{{{x^2}}}\\&= \frac{{62.5}}{{{5^2}}}\\&=\frac{{62.5}}{{25}}\\&= 2.5\\\end{aligned}$

The height of the box is $y = 2.5{\text{ in}}.$

The surface area of the box can be calculated as follows,

$\begin{aligned}{\text{Surface area}} &= {\left( 5 \right)^2} + 4\left( {5 \times 2.5} \right)\\&= 25 + 4\left( {12.5} \right)\\&= 25 + 50\\&= 75{\text{ i}}{{\text{n}}^2}\\\end{aligned}$

The dimensions of minimize surface area are $\boxed{5{\text{ in}}}{\text{ and }}\boxed{{\text{2}}{\text{.5 in}}}$ and the minimum surface area is $\boxed{75{\text{ i}}{{\text{n}}^2}}.$

Learn more:

1. Learn more about inverse of the functionhttps://.

2. Learn more about equation of circle .

3. Learn more about range and domain of the function

Answer details:

Grade: High School

Subject: Mathematics

Chapter: Application of Derivatives

Keywords: Drum, tight containers, open top, square base, volume of 62.5 inches cubed, rectangular box, minimum surface area, dimensions, designing.

Answer from: Quest

answer: 177/8

step-by-step explanation:

Solve for x i’m having trouble finding out the answer

Answer from: Quest

"it's not the heat, it's the humidity". that's a partly valid phrase you may have heard in the summer, but it's actually both. the heat index, also known as the apparent temperature, is what the temperature feels like to the human body when relative humidity is combined with the air temperature. this has important considerations for the human body's comfort. when the body gets too hot, it begins to perspire or sweat to cool itself off. if the perspiration is not able to evaporate, the body cannot regulate its temperature. evaporation is a cooling process. when perspiration is evaporated off the body, it effectively reduces the body's temperature. when the atmospheric moisture content (i.e. relative humidity) is high, the rate of perspiration from the body decreases. in other words, the human body feels warmer in humid conditions. the opposite is true when the relative humidity decreases because the rate of perspiration increases. the body actually feels cooler in arid conditions. there is direct relationship between the air temperature and relative humidity and the heat index, meaning as the air temperature and relative humidity increase (decrease), the heat index increases (decreases).

step-by-step explanation:

75% of the temperatures are below what value? how do you know? 75% of the temperatures are above w

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Drum tight containers is designing an open-top, square-based, rectangular box that will have a volum...