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Water is leaking out of a large barrel at a rate proportional to the square rooot of the depth of the water at that time. If the

sdas [7]

Answer:

It will take about 35.49 hours for the water to leak out of the barrel.

Step-by-step explanation:

Let $y(t)$ be the depth of water in the barrel at time $t$ , where $y$ is measured in inches and $t$ in hours.

We know that water is leaking out of a large barrel at a rate proportional to the square root of the depth of the water at that time. We then have that

$\frac{dy}{dt}=-k\sqrt{y}$

where $k$ is a constant of proportionality.

Separation of variables is a common method for solving differential equations. To solve the above differential equation you must:

Multiply by $\frac{1}{\sqrt{y}}$

$\frac{1}{\sqrt{y}}\frac{dy}{dt}=-k$

Multiply by $dt$

$\frac{1}{\sqrt{y}}\cdot dy=-k\cdot dt$

Take integral

$\int \frac{1}{\sqrt{y}}\cdot dy=\int-k\cdot dt$

Integrate

$2\sqrt{y}=-kt+C$

Isolate $y$

$y(t)=(\frac{C}{2} -\frac{k}{2}t)^2$

We know that the water level starts at 36, this means $y(0)=36$ . We use this information to find the value of $C$ .

$36=(\frac{C}{2} -\frac{k}{2}(0))^2\\C=12$

$y(t)=(\frac{12}{2} -\frac{k}{2}t)^2\\\\y(t)=(6 -\frac{k}{2}t)^2$

At t = 1, y = 34

$34=(6 -\frac{k}{2}(1))^2\\k=12-2\sqrt{34}$

So our formula for the depth of water in the barrel is

$y(t)=(6 -\frac{12-2\sqrt{34}}{2}t)^2\\\\y(t)=\left(6-\left(6-\sqrt{34}\right)t\right)^2\\$

To find the time, $t$ , at which all the water leaks out of the barrel, we solve the equation

$\left(6-\left(6-\sqrt{34}\right)t\right)^2=0\\\\t=3\left(6+\sqrt{34}\right)\approx 35.49$

Thus, it will take about 35.49 hours for the water to leak out of the barrel.

5 0

3 years ago

Evaluate xy-x; use c =4, and y =4

Ilya [14]

Answer:

12

Step-by-step explanation:

You would substitute

4(4)-4

then simplify

16-4

then subtract

12

4 0

3 years ago

The price for 7/8 of a pound of almonds is $6.65 . What is the cost of 3 and 1/2 pounds at the same rate?

Vlada [557]

The cost of 3 and 1/2 pounds of almond at the same rate is $26.60.

<h3>What is a fraction?</h3>

A fraction is simply a piece of a whole. The number is represented mathematically as a quotient where the numerator and denominator are split. In the illustration given as m/n, it should be noted that m is the numerator and n is the denominator.

Since the price for 7/8 of a pound of almonds is $6.65, the fraction per pound will be done by dividing the amount for a pound by 7/8.

= $6.65 ÷ 7/8

= $7.6

The cost of 3 and 1/2 pounds at the same rate is:

= 3 1/2 × $7.6

= $26.60

In conclusion, the cost is $26.60.

Learn more about fractions on:

brainly.com/question/78672

#SPJ1

5 0

1 year ago

I need help with number 4 if you know it

Svetradugi [14.3K]

Answer:

the answer is 5500 but if not put 6000$

3 0

2 years ago

1. A café menu has the following choices of cheese and toppings for scrambled

Vinil7 [7]

Answer:

the top answer in your picture is the right answer

Step-by-step explanation:

8 0

3 years ago