The revenue in dollars from the sale of concert tickets at x dollars each is given by the function R(x)=46,400x-800x^2. Find the difference quotient when x=27 a h=0.1, and when x=31and h=o.1. Intepret your answers
To find the difference quotient, we need to use the formula:
Difference Quotient = (R(x + h) - R(x)) / h
Let's find the difference quotient when x = 27 and h = 0.1:
R(x + h) = R(27 + 0.1) = R(27.1)
R(x) = R(27)
Using the revenue function, we can calculate R(27.1):
R(27.1) = 46,400(27.1) - 800(27.1)^2
R(27) = 46,400(27) - 800(27)^2
Now, substitute these values into the difference quotient formula:
Difference Quotient = (R(27.1) - R(27)) / 0.1
Calculate:
Difference Quotient = [(46,400(27.1) - 800(27.1)^2) - (46,400(27) - 800(27)^2)] / 0.1
Simplify:
Difference Quotient = (1,257,040 - 629,760) / 0.1
Difference Quotient = 627,280 / 0.1
Difference Quotient = 6,272,800
Now let's find the difference quotient when x = 31 and h = 0.1:
R(x + h) = R(31 + 0.1) = R(31.1)
R(x) = R(31)
Using the revenue function, we can calculate R(31.1):
R(31.1) = 46,400(31.1) - 800(31.1)^2
R(31) = 46,400(31) - 800(31)^2
Now, substitute these values into the difference quotient formula:
Difference Quotient = (R(31.1) - R(31)) / 0.1
Calculate:
Difference Quotient = [(46,400(31.1) - 800(31.1)^2) - (46,400(31) - 800(31)^2)] / 0.1
Simplify:
Difference Quotient = (1,442,240 - 721,920) / 0.1
Difference Quotient = 720,320 / 0.1
Difference Quotient = 7,203,200
Interpretation:
The difference quotient represents the average rate of change of the revenue function with respect to the change in the ticket price (x), over a small interval (h).
When x = 27 and h = 0.1, the difference quotient is 6,272,800. This means that for every $0.1 increase in the ticket price from $27, the revenue increases by an average rate of $6,272,800.
When x = 31 and h = 0.1, the difference quotient is 7,203,200. This means that for every $0.1 increase in the ticket price from $31, the revenue increases by an average rate of $7,203,200.
To find the difference quotient, we need to substitute the given values into the function and calculate the difference between two outputs, divided by the difference in inputs. The general formula for the difference quotient is:
Difference Quotient = (f(x + h) - f(x)) / h
Let's calculate the difference quotient for both cases:
1. When x = 27 and h = 0.1:
1.1. Calculate f(x + h):
f(27 + 0.1) = R(27.1) = 46,400(27.1) - 800(27.1)^2
1.2. Calculate f(x):
f(27) = R(27) = 46,400(27) - 800(27)^2
1.3. Calculate the difference quotient:
Difference Quotient = (f(27.1) - f(27)) / 0.1
2. When x = 31 and h = 0.1:
2.1. Calculate f(x + h):
f(31 + 0.1) = R(31.1) = 46,400(31.1) - 800(31.1)^2
2.2. Calculate f(x):
f(31) = R(31) = 46,400(31) - 800(31)^2
2.3. Calculate the difference quotient:
Difference Quotient = (f(31.1) - f(31)) / 0.1
The interpretation of the difference quotient is that it represents the average rate of change of the function over the given interval (h). In this case, the difference quotient will approximate the average rate of change of the revenue function for a small change in ticket price.
By calculating the specific values, we can determine the exact average rate of change of the revenue function at x = 27 and x = 31.
well, just plug and chug:
R(x+h)-R(x)
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h