Mathematics, 26.10.2020 16:40 bethdove9466

The volume of a rectangular box with a square base remains constant at 1000 cm3 as the area of the base increases at a rate of 4 cm2 /sec. Find the rate at which the height of the box is decreasing when each side of the base is 12 cm long.

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The rate at which the height of the box is decreasing when each side of the base is 12 centimeters long is 0.193 centimeters per second.

Step-by-step explanation:

We can represent the volume of the rectangular box with a square base (), measured in cubic centimeters, by means of this formula:

$V = A\cdot h$ (Eq. 1)

Where:

- Area of the base of the box, measured in centimeters.

- Height of the rectangular box, measured in centimeters.

Given that volume remains constant, we clear the height of the box as follows:

$h = V\cdot A^{-1}$ (Eq. 1b)

By Differential Calculus, we get the expression for the rate of change of the height:

$\frac{dh}{dt} = - V\cdot A^{-2}\cdot \frac{dA}{dt}$ (Eq. 2)

Where:

$\frac{dA}{dt}$ - Rate of change of the area of the base in time, measured in square centimeters per second.

$\frac{dh}{dt}$ - Rate of change of the height of the box in time, measured in centimeters per second.

If we know that $V = 1000\,cm^{3}$ , $A = 144\,cm^{2}$ and $\frac{dA}{dt} = 4\,\frac{cm^{2}}{s}$ , then the rate of change of the area of the base in time is:

$\frac{dh}{dt} = -(1000\,cm^{3})\cdot (144\,cm^{2})^{-2} \cdot \left(4\,\frac{cm^{2}}{s} \right)$

$\frac{dh}{dt} = -0.193\,\frac{cm}{s}$

The rate at which the height of the box is decreasing when each side of the base is 12 centimeters long is 0.193 centimeters per second.

Answer from: Quest

Hello!

the correct option is:

b. a value of the variable that makes the polynomial equal to zero.

why?

the root or roots of a polynomial are the values of the variable that makes the polynomial equal to 0. we can find the roots of a polynomial making the expression equal to 0 and then, isolating the variable (it it has just one variable).

for example, given the following polynomial:

$3x-6$

the root or zero of the polynomial, can be found by making the expression equal to 0, and then, isolating the variable, so, finding the root we have:

$3x-6=0$

$3x=6$

$x=\frac{6}{3}=2$

therefore, we have that the root of the polynomial is equal to 2.

we must remember that the number of roots that a polynomial can have, will depend on the degree of the polynomial, a first degree polynomial have will have one single root, a second degree polynomial (quadratic expression) may have two roots and so.

hence the correct option is:

b. a value of the variable that makes the polynomial equal to zero.

have a nice day!

Answer from: Quest

the option best identifies the textual evidence that supports this conclusion is c.) when an argument is over, how many weighty reasons does a man recollect which his heat and violence made him utterly forget? in order to recognize textual evidence is to identify the topic sentence where the author states the main idea. textual evidence is what supports the topic sentence or main idea of a passage.

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