Find the range of the function:

A storage box with a square base must have a volume of 90 cubic centimeters. the top and bottom cost $0.60 per square centimeter

Paul [167]

Answer-

For side length of 3.56 cm and height of 7.10 cm the cost will be minimum.

Solution-

Let us assume that,

x represents the length of the sides of the square base,

y represent the height.

Given the volume of the box is 90 cm³, so

$\Rightarrow V=90\\\\\Rightarrow x^2\times y=90\\\\\Rightarrow y=\dfrac{90}{x^2}$

As the top and bottom cost $0.60 per cm² and the sides cost $0.30 per cm². Total cost C will be,

$C=\text{cost for top and bottom}+\text{cost for rest 4 sides}\\\\=(2x^2\times 0.6)+(4xy\times 0.3)\\\\=(2x^2\times 0.6)+(4x\times \frac{90}{x^2}\times 0.3)\\\\=1.2x^2+ \dfrac{108}{x}$

Then,

$C'=\dfrac{d}{dx}(1.2x^2+ \dfrac{108}{x})=2.4x-\dfrac{108}\\\\C''=\dfrac{d^2}{dx^2}(1.2x^2+ \dfrac{108}{x})=2.4+\dfrac{216}{x^3}$

As C'' has all positive terms so, for every positive value of x (as length can't be negative), C'' is positive.

So, for minima C' = 0

$\Rightarrow 2.4x-\dfrac{108}{x^2}=0\\\\\Rightarrow 2.4x=\dfrac{108}{x^2}\\\\\Rightarrow x^3=\dfrac{108}{2.4}=45\\\\\Rightarrow x=3.56$

Then,

$y=\dfrac{90}{x^2}$

$y=\dfrac{90}{3.56^2}$

$y=7.10$

Therefore, for side length of 3.56 cm and height of 7.10 cm the cost will be minimum.

4 0

4 years ago

Read 2 more answers

Lydia is at a coffee shop and knows she can spend no more than $65 before tax. She sees this price list in the coffee shop. Writ

Artyom0805 [142]

Price chart :

Item Price per pound

Dark Roast Coffee $7.50

Pumpkin Spice Coffee $10.50

Breakfast Tea $23.50

Answer:

Step-by-step explanation:

4

Maximum amount to spend ≤ 65

Cost of dark roast coffee = $7.50

2 dark roast coffee= $7.50 * 2 = $15

Pumpkin Spice + dark roast ≤ amount to spend

Let pumpkin spice = x

Cost of pumpkin spice = $10.50

Dark Roast Coffee = y

10.50x + 2* 7.50 ≤ 65

10.50x + 15 ≤ 65

10.50x ≤ 65 - 15

10.50x ≤50

x ≤50 / 10.50

x ≤ 4.76

Number of pumpkin spice coffee that can be purchased is 4.

5 0

3 years ago

A nest if ants initially contains 500 individuals. The population is increasing by 12% each week. How many ants will there be af

allochka39001 [22]

Based on the given values above, here is the solution to find if how many ants there will be after 10 weeks and 20 weeks.
Since the population increases by 12%. we need to get 12% of the population every week.
500 x .12 = 60.
1st week = 560
2nd week = 627.2
3rd week = 702. 46
4th week = 786.76
5th week = 881. 17
6th week = 986.91
7th week= 1,105.34
8th week= 1,237.98
9th week= 1, 386.54
10th week = 1, 552.92
11th week = 1,739.27
12th week = 1,947.98
13th week= 2,181.74
14th week=2, 443.55
15th week = 2,736.76
16th week = 3, 065. 17
17th week = 3,432.99
18th week= 3,844.95
19th week= 4,306.34
20th week = 4,823.10
Therefore, after 10 weeks, there will be 1,553 ants and after 20 weeks, there will be 4,823 ants.

8 0

3 years ago

Find an expression which represents the sum of (-5x + 10) and (-7x + 9) in simplest terms.

Brrunno [24]

Answer:

-12x+19

Step-by-step explanation:

Combine like terms first soooo

-5x-7x=-12x

10+9=19

-12x+19 is your answer

3 0

3 years ago

At the Cubs/White Sox baseball game on Sunday night, 65% of the fans were men. Another 25% were women. Of the remainder, 40% wer

strojnjashka [21]

Ok, 65% of the attendants were men, 25% were women.

That leaves us with 10% remaining.

From those 10% .................The problem tells us that 40% are boys

So 10% * ( 0.4 ) = 4% ..............This means that 4% of the total attendants were boys.

So the distribution is

65%........men
25%........women
4%..........boys
6%...........girls

640 boys attended the game.....> this means that 4% of the total amount of people is 640.

So the number of persons that attended were

T= 100% *(640)) / 4% = 16000 people

5 0

3 years ago