A coal-burning electrical generating plant emits sulfur dioxide into the surrounding air. The concentration C(x), in parts per million,is approximately given by the function:
C(x)= 0.5/x^2,
where x is the distance away from the plant in miles. The instantaneous rate of change of sulfur dioxide concentration:
A) 5 miles from the plant=____
B) 8 miles from the plant=____
A) 5 miles from the plant = 0.5/5^2 = 0.5/25 = 0.02 parts per million per mile.
B) 8 miles from the plant = 0.5/8^2 = 0.5/64 = 0.0078125 parts per million per mile.
And remember, if you have a powerful sense of smell, you might want to stay even farther away!
To find the instantaneous rate of change of sulfur dioxide concentration, we need to take the derivative of the concentration function with respect to x.
Differentiating the function C(x) = 0.5/x^2:
C'(x) = -1.0/x^3
The negative sign indicates that the concentration is decreasing as the distance from the plant increases.
Now, we can find the instantaneous rate of change at specific distances.
A) 5 miles from the plant:
Substituting x = 5 into the derivative equation:
C'(5) = -1.0/5^3
= -1/125
= -0.008
Therefore, the instantaneous rate of change of sulfur dioxide concentration 5 miles from the plant is -0.008 parts per million per mile.
B) 8 miles from the plant:
Substituting x = 8 into the derivative equation:
C'(8) = -1.0/8^3
= -1/512
≈ -0.002
Therefore, the instantaneous rate of change of sulfur dioxide concentration 8 miles from the plant is approximately -0.002 parts per million per mile.
To find the instantaneous rate of change of the sulfur dioxide concentration at a specific distance from the plant, we need to calculate the derivative of the concentration function, C(x). The derivative will give us the rate of change of the concentration with respect to x.
Let's start by finding the derivative of C(x):
C'(x) = d/dx (0.5/x^2)
To differentiate this function, we use the power rule, which states that for a function of the form f(x) = k/x^n, the derivative is given by f'(x) = -kn/x^(n+1).
Applying the power rule to C(x), we have:
C'(x) = -0.5 * (-2) / x^(2+1)
= 1/x^3
Now we have the derivative of the concentration function, C(x), which represents the instantaneous rate of change of the sulfur dioxide concentration at any distance x from the plant.
To find the instantaneous rate of change at a specific distance, we substitute that distance into the derivative function C'(x).
A) 5 miles from the plant:
C'(5) = 1/5^3
= 1/125
= 0.008
Therefore, the instantaneous rate of change of sulfur dioxide concentration 5 miles from the plant is approximately 0.008 parts per million per mile.
B) 8 miles from the plant:
C'(8) = 1/8^3
= 1/512
≈ 0.00195
Therefore, the instantaneous rate of change of sulfur dioxide concentration 8 miles from the plant is approximately 0.00195 parts per million per mile.
C ' (x) = - x^-3 or -1/x^3
when x = 5, C ' (5) = -1/125
when x = 8 , .......