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LuckyWell [14K]
3 years ago
9

an inverted conical water tank with a height of 20 ft and a radius of 8 ft is drained through a hole in the vertex (bottom) at a

rate of 4 ft^3/sec. Find the rate of change of the water depth when the water depth is 10 ft.

Mathematics
1 answer:
viktelen [127]3 years ago
4 0

Answer:

the rate of change of the water depth when the water depth is 10 ft is;  \mathbf{\dfrac{dh}{dt}  = \dfrac{-25}{100  \pi} \  \ ft/s}

Step-by-step explanation:

Given that:

the inverted conical water tank with a height of 20 ft and a radius of 8 ft  is drained through a hole in the vertex (bottom) at a rate of 4 ft^3/sec.

We are meant to find the  rate of change of the water depth when the water depth is 10 ft.

The diagrammatic expression below clearly interprets the question.

From the image below, assuming h = the depth of the tank at  a time t and r = radius of the cone shaped at a time t

Then the similar triangles  ΔOCD and ΔOAB is as follows:

\dfrac{h}{r}= \dfrac{20}{8}    ( similar triangle property)

\dfrac{h}{r}= \dfrac{5}{2}

\dfrac{h}{r}= 2.5

h = 2.5r

r = \dfrac{h}{2.5}

The volume of the water in the tank is represented by the equation:

V = \dfrac{1}{3} \pi r^2 h

V = \dfrac{1}{3} \pi (\dfrac{h^2}{6.25}) h

V = \dfrac{1}{18.75} \pi \ h^3

The rate of change of the water depth  is :

\dfrac{dv}{dt}= \dfrac{\pi r^2}{6.25}\  \dfrac{dh}{dt}

Since the water is drained  through a hole in the vertex (bottom) at a rate of 4 ft^3/sec

Then,

\dfrac{dv}{dt}= - 4  \ ft^3/sec

Therefore,

-4 = \dfrac{\pi r^2}{6.25}\  \dfrac{dh}{dt}

the rate of change of the water at depth h = 10 ft is:

-4 = \dfrac{ 100 \ \pi }{6.25}\  \dfrac{dh}{dt}

100 \pi \dfrac{dh}{dt}  = -4 \times 6.25

100  \pi \dfrac{dh}{dt}  = -25

\dfrac{dh}{dt}  = \dfrac{-25}{100  \pi}

Thus, the rate of change of the water depth when the water depth is 10 ft is;  \mathtt{\dfrac{dh}{dt}  = \dfrac{-25}{100  \pi} \  \ ft/s}

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Which linear equation has no solution?
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Option A

Step-by-step explanation:

Given:

  • a. 3x-5= 3x + 5
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  • Which one of the linear equations have no solution.

Solution:

a)  3x-5= 3x + 5

Add 5 to both sides

3x-5= 3x + 5

3x - 5 + 5 = 3x + 5 + 5

Simplify

(Add the numbers)

3x - 5 + 5 = 3x + 5 + 5

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(Add the numbers)

3x  = 3x + 5 + 5

3x = 3x + 10

Subtract 3x from both sides

3x = 3x + 10

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Simplify

(Combine like terms)

3x -3x = 3x + 10 - 3

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(Combine like terms)

0 = 3x + 10 - 3

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The input is a contradiction: it has no solutions

b)  3x-5= 3x - 5

Since both sides equal, there are infinitely many solutions.

c)  3x - 5 = 2x+5

Add 5 to both sides

3x = 2x + 5 + 5

Simplify  2x + 5 + 5 to 2x + 10

3x = 2x + 10

Subtract 2x from both sides

3x - 2x = 10

Simplify 3x - 2x to x.

x = 10

d) 3x-5 = 2x + 10

Add 5 to both sides

3x = 2x + 10 + 5

Simplify 2x + 10 + 5 to 2x + 15

3x = 2x + 15

Subtract 2x from both sides

3x - 2x = 15

Simplify 3x -2x to x.

x = 15

--------------------------------------------

Answer:

As you can see all c and d both have solutions, eliminating them as options. Option B has infinite solutions leaving Option A which has no solutions.

Therefore, <u><em>Option A</em></u> is the linear equation that has no solution.

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