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A rectangle box with a square base and no top needs to be made using 300ft^2 of material. Find the dimensions of the box with th

Yuri [45]

The dimensions of the box are 10 ft and 5 ft

The maximum volume is 500 ft³

Step-by-step explanation:

A rectangle box with

A square base and no top
It needs to be made using 300 ft² of material
It has greatest volume

Surface area of a box without top (SA) = perimeter of base × height + area of the base

Volume of a box (V) = base area × height

Assume that the length of the side of the square base is x and the height of the box is y

∵ It needs to be made using 300 ft² of material

∴ The surface area of the box is 300 ft²

∵ Its base is a square of side length x ft

∴ Perimeter of the base = 4 × x = 4 x

∴ Area of the base = x²

∵ The height of the box = y ft

∵ SA = perimeter of base × height + area of the base

∵ SA = (4x)(y) + x²

∴ SA = 4xy + x²

∵ SA of the box = 300 ft²

- Equate the two expressions of SA

∴ 4xy + x² = 300

Now let us find y in terms of x

- Subtract x² from both sides

∴ 4xy = 300 - x²

- Divide each term by 4x to find y

∴ $y=\frac{75}{x}-\frac{1}{4}x$

∵ V = area of the base × height

∴ V = x² × y = x²y

- Substitute y by the equation of it above

∴ $V=x^{2}(\frac{75}{x}-\frac{1}{4}x)$

∴ $V=75x-\frac{1}{4}x^{3}$

∵ The volume of the box is greatest

- That means differentiate V and equate it by 0

∵ $\frac{dV}{dx}=75-\frac{3}{4}x^{2}$

∵ $\frac{dV}{dx}=0$ ⇒ greatest volume

∴ $75-\frac{3}{4}x^{2}=0$

- Subtract 75 from both sides

∴ $-\frac{3}{4}x^{2}=-75$

- Divide both sides by $-\frac{3}{4}$

∴ x² = 100

- Take √ for both sides

∴ x = 10

Substitute the value of x in the equation of y

∵ $y=\frac{75}{10}-\frac{1}{4}(10)$

∴ y = 5

The dimensions of the box are 10 ft and 5 ft

∵ $V=75x-\frac{1}{4}x^{3}$

∵ x = 10

∴ $V=75(10)-\frac{1}{4}(10)^{3}$

∴ $V=750-\frac{1}{4}(1000)$

∴ V = 750 - 250

∴ V = 500 ft³

The maximum volume is 500 ft³

Learn more:

You can learn more about the volume in brainly.com/question/6443737

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6 0

3 years ago

There are 39 chips number from 1 to 39 placed in a barrel one chip is randomly pulled from the barrel what is the probability th

Alex Ar [27]

P (number on chip is 18 or higher) = 22/39

7 0

3 years ago

URGENT will give brilliant mark For the reaction 2Na + Cl2 ------ 2NaCl, how many grams of sodium chloride can be produced from

SVETLANKA909090 [29]

Answer:

825 g

Step-by-step explanation:

Here's my work:

500g Cl2 l 1mol Cl2 l 2 mol NaCl l 58.5 g NaCl

l 71 g Cl2 l 1 mol Cl2 l 1 mol NaCl

= 823.94 - the rounding can be a little off

The first thing you have to do is convert the 500 g of Cl2 into moles. So you get the mass of Cl which is 35.5, but Cl is diatomic by itself so you multiply 35.5 x 2 and get 71. now on the third column you are comparing the mol ratios of the equation. The equation is 2Na + Cl2 ---> 2NaCl, so you see that there is no coefficient in front of Cl2 so that means there is 1 mol, and then you see that in the product there is a coefficient of 2 in front of NaCl so that means there is 2 moles. Then finally you have to convert the moles backs into grams where you just add the masses of Na and Cl . You multiply across the top and then divide that number by the bottom row.

6 0

3 years ago

PLZ HELP ME PLZZZ 15 PTS Mr. Rodriguez asked 60 randomly chosen students from each grade level at Willowbrook School about their

vredina [299]

12/60 students chose science fiction

Approximately x/150 students prefer sf

60x = 1800

x = 30

30/150 students are assumed to prefer sf

30/150 = x/100

150x = 3000

x = 20

20/100 students are likely to prefer sf

Mr. Rodriguez made a reasonable estimate for the approximate percentage of students that prefer science fiction, because if 12/60 is equivalent to 30/150 which refers to the number of students who can be assumed to prefer science Fiction out of the whole school. Considering we need to identify what 30/150 as a percentage is, we can reduce it down to 1/5 to make I easier, then divide 1 by 5 to get .2
.2 as a percentage is 20%, so his inference was indeed reasonable.

(♥ω♥*)Brainliest Please(♥ω♥*)

8 0

3 years ago

Read 2 more answers

After the booster club sold 40 hotdogs at a football game, it had $90 in profit. After the next game, it had sold a total of 80

Ainat [17]

We can model the equation:y = m x + b, where y is the total profit and x is the number of hotdogs sold.The system of equations is:
90 = 40 m + b210 = 80 m + b---------------------b = 90 - 40 m210 = 80 m + 90 - 40 m210 - 90 = 40 m120 = 40 mm = 120 : 40m = 3b = 90 - 40*3b = 90 - 120b = - 30Answer: The equation is y = 3 x - 30

5 0

3 years ago