The volume V of a gas varies inversly as pressure P is exerted. If V is 3.5 liters when P is 5 atmospheres, find V when P is 8 atmospheres.

Answer: 2.188 liters

2) If y varies jointly as x and the cube of z, and y=378 when x=4 and z=3, find y when x=9 and z=2.

Answer: y=252

yes for both

length of 12:19 m in kilometers.

To find the volume V when pressure P is 8 atmospheres, we need to use the concept of inverse variation. Inverse variation means that when one variable increases, the other variable decreases in the same proportion, and vice versa.

In this case, we know that the volume V varies inversely with pressure P, so we can set up the equation:

V ∝ 1/P

To find the constant of variation, we can use the initial values given in the problem. When V is 3.5 liters, P is 5 atmospheres. Plugging these values into the equation, we get:

3.5 ∝ 1/5

To isolate V, we can cross-multiply:

3.5 * 5 = 1 * V
17.5 = V

So, when P is 5 atmospheres, V is 17.5 liters.

Now, we can use this information to find V when P is 8 atmospheres. Again, we set up the equation:

V ∝ 1/P

Plugging in the new value, we have:

V ∝ 1/8

To find V, we need to multiply this value by the constant of variation. Using the value we found earlier, we have:

V = (1/8) * 17.5
V = 2.188 liters

Therefore, when P is 8 atmospheres, the volume V is 2.188 liters.

Now let's move on to the second question:

To find y when x is 9 and z is 2, we are given that y varies jointly as x and the cube of z. Joint variation means that a variable varies directly with the product of two or more other variables. In this case, we have:

y ∝ x * z^3

To find the constant of variation, we can use the initial values given in the problem. When y is 378, x is 4, and z is 3. Plugging these values into the equation, we get:

378 ∝ 4 * 3^3

Simplifying, we have:

378 ∝ 4 * 27
378 ∝ 108

To isolate y, we can cross-multiply:

378 * 1 = 108 * y
y = 378/108
y = 3.5

So, when x is 4 and z is 3, y is 3.5.

Now, we can use this information to find y when x is 9 and z is 2. Again, we set up the equation:

y ∝ x * z^3

Plugging in the new values, we have:

y ∝ 9 * 2^3

Simplifying, we have:

y ∝ 9 * 8
y ∝ 72

Therefore, when x is 9 and z is 2, y is 72.