Mathematics, 29.07.2019 11:30 nickname0097

At a sand and gravel plant, sand is falling off a conveyor and onto a conical pile at a rate of 10 cubic feet per minute. the diameter of the base of the cone is approximately three times the altitude. at what rate is the height of the pile changing when the pile is 12 feet high?

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Answer from: mimithurmond03

The rate of the height of the pile changing when the pile is 12 feet high is 101.828.

Step-by-step explanation:

Sand is falling off a conveyor and onto a conical pile at a rate of 10 cubic feet per minute.

Therefore, if the volume is denoted by ''V" then the rate of change of the volume is 10.

So,

$\bold{\dfrac{dV}{dt}=10}$

The diameter of the base of the cone is approximately three times the altitude.

Thus,

$D=3h\\2r=3h\\r=\dfrac{3h}{2}$

Now,

The expression for finding the volume of the cone is formulated as:

$\begin{aligned} V=\dfrac{1}{3} \pi r^2h\\V=\dfrac{1}{3} \pi (\dfrac{3h}{2})^2h\\V=\dfrac{3\pi h^3}{4}\end{aligned}$

Now,

Differentiate the above equation with respect to time.

$\dfrac{dV}{dt}=\dfrac{d}{dt}\left[\dfrac{3\pi h^3}{4}\right]\\\dfrac{dV}{dt}=\left[\dfrac{9\pi h^2}{4}\right]\dfrac{dh}{dt}$

Now,

Rate is the height of the pile changing is $\dfrac{dh}{dt}$ .

Height is given in the question that is 12 feet.

Therefore,

$\dfrac{dV}{dt}=\left[\dfrac{9\pi (12)^2}{4}\right]\dfrac{dh}{dt}\\10=324 \pi \dfrac{dh}{dt}\\\dfrac{dh}{dt}=32.4 \pi\\\dfrac{dh}{dt}=101.828$

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Answer from: croxy0514

1. You have that the rate is10 ft³/min. Then:

dV/dt=10

2. The formula for calculate the volume of a cone, is:

V=1/3(πr²h)

"r" is the radius and "h" is the height.

3. The diameter of the base of the cone is approximately 3 times the altitude. Then, the radius is:

r=diameter/2

diameter=3h

r=3h/2

4. When you susbstitute r=3h/2 into the formula V=πr²h/3, you have:

V=1/3(πr²h)
V=1/3(π(3h/2)²(h)
V=1/3(π9h²/4)(h)
V=9πh³/12

5. Therefore:

dV/dt=(9πh²/4)dh/dt

h=12

dV/dt=10

6. When you substitute the values of dV/dt and h into dV/dt=(9π(12)²/4)dh/dt, you have:

dV/dt=(9π(12)²/4)dh/dt
10=(1017.876)

7. Finally, you obtain:

dh/dt=10/1017.876
dh/dt=9.82x10^-3 ft/min

Answer from: Quest

0.16 or 16%

step-by-step explanation:

lets take a coin for example.

if you toss it twice, it has a 1/4 chance of landing on heads both times because 1/2 * 1/2 is 1/4.

with the cards, the chance of picking dog is 0.4. having two people pick one is like picking a card twice in a row. so 0.4 * 0.4 is 0.16, or 16%

Answer from: Quest

the results from a pre-test for students for the year 2000 and the year 2010 are illustrated in the box plot. what do these results tell us about how students performed on the 29 question pre-test for the two years?

if we compare only the lowest and highest scores between the two years, we might conclude that the students in 2010 did better than the students in 2010. this conclusion seems to follow since the lowest score of 8 in 2010 is greater in value than the lowest score of 6 in 2000. also, the highest score of 28 in 2010 is greater in value than the highest score of 27 in 2000.

but the box portion of the illustration gives us more detailed information. the middle bar in each box shows us that the median score of 20 in 2000 is greater in value than the median score of 17 in 2010. further, we note that the box and whiskers divide the illustration into four pieces. each of these four pieces represents the same portion of students. so, the upper half of the students in 2000 scored in the same score range as the upper one-fourth of the students in 2010, see the illustration at a score of 20.

by considering the upper one-fourth, upper half, and upper three-fourths instead of just the lowest and highest scores, we would conclude that the students as a whole did much better in 2000 than in 2010. we would conclude that as a whole the students in 2010 are less prepared than the students in 2000.

in this section, we discuss box-and-whisker plots and the five key values used in constructing a box-and-whisker plot. the key values are called a five-number summary, which consists of the minimum, first quartile, median, third quartile, and maximum.

step-by-step explanation:

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