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Gelneren [198K]
3 years ago
7

What is the maximum volume in cubic inches of an open box to be made from a 12-inch by 16-inch piece of cardboard by cutting out

squares of equal sides from the four corners and bending up the sides? Your work must include a statement of the function and its derivative. Give one decimal place in your final answer. (10 points)
Mathematics
1 answer:
Romashka-Z-Leto [24]3 years ago
5 0

If you start with a 12x16 rectangle and cut square with side length x, when you bend the sides you'll have an inner rectangle with sides 12-2x and 16-2x, and a height of x.

So, the volume will be given by the product of the dimensions, i.e.

(12-2x)(16-2x)x = 4x^3-56x^2+192x

The derivative of this function is

12x^2-112x+192

and it equals zero if and only if

12x^2-112x+192=0 \iff x = \dfrac{14\pm 2\sqrt{13}}{3}

If we evaluate the volume function at these points, we have

f\left(\dfrac{14-2\sqrt{13}}{3}\right) = \dfrac{64}{27}(35+13\sqrt{13})> f\left(\dfrac{14-2\sqrt{13}}{3}\right) = -\dfrac{64}{27}(13\sqrt{13}-35)

So, the maximum volume is given if you cut a square with side length

x=\dfrac{14-2\sqrt{13}}{3}\approx 2.7

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aleksandr82 [10.1K]

12. 1.625 [terminating]; 13. 0.83 [bar notation over 3 (repeating)]; 14. 900 cm = 9 m; 15. 0.23 cm = 2.3 mm

Repeating decimals are parts of decimals that have repetitive digits; terminating decimals are decimals whose digits end.

Whether you are using Metric or Imperial, you have to determine whether you are going from a small unit to a big unit or vice versa. Then perform your operation. So, in exercise 14, the smaller unit is centimeters, so you would be going from big to small. Exercise 15 has you going from small to big.

There are centimeters in one meter, so multiply 9 by to get 900 centimeters.

There are 10 millimeters in one centimeter, so divide 2.3 by 10 simply by moving the decimal point ONCE to the left [Power of 10].

small to BIG → Division

BIG to small → Multiplication

I am joyous to assist you anytime.

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3 years ago
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Gemiola [76]

Answer:

24

Step-by-step explanation:

2:6

1:3

24:72

8 0
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Answer:

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2 years ago
Jimmy is working at a factory where they make cars and trucks in the ratio five to four if the factory makes 100 trucks how many
-BARSIC- [3]

Answer:

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Step-by-step explanation:

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4 0
2 years ago
4 times a number is 16​
andrey2020 [161]

Answer:

4 times 4 equals 16

Step-by-step explanation:

4 times 1= 4

4 times 2= 8

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4 times 4= 16

you add 4 after each result.

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