Boyle's Law states that when a sample of gas is compressed at a constant temperature, the pressure P and volume V satisfy the equation P V = C , where C is a constant. Suppose that at a certain instant the volume is 800 cm 3 , the pressure is 160 kPa , and the pressure is increasing at a rate of 40 kPa/min . At what rate is the volume decreasing at this instant?

The volume decreasing at the rate of .

Step-by-step explanation:

We need to find at what rate is the volume decreasing , when , , and .

We know that when a sample of gas is compressed at a constant temperature, the pressure P and volume V satisfy the equation , where C is a constant.

So, we can differentiate this equation with respect the time. For this we use  the Chain Rule

and the Product rule

to differentiate both sides with respect to time.

Next, we substitute the given values

and we solve for

y= -

use the power rule  

( a b ) n = a n b n

to distribute the exponent.

apply the product rule to  

3 x 2 y 3 . ( 3 x 2 ) 2 ( y 3 ) 2

apply the product rule to  

3 x 2 . 3 2 ( x 2 ) 2 ( y 3 ) 2

raise  

3

to the power of  

2 . 9 ( x 2 ) 2 ( y 3 ) 2 multiply the exponents in   ( x 2 ) 2

.

apply the power rule and multiply exponents,  

( a m ) n = a m n . 9 x 2 ⋅ 2 ( y 3 ) 2

multiply  

2   by   2 . 9 x 4 ( y 3 ) 2

multiply the exponents in  

( y 3 ) 2 .

apply the power rule and multiply exponents,  

( a m ) n = a m n . 9 x 4 y 3

⋅

2

multiply  

3   by   2 . 9 x 4 y 6

step-by-step explanation: