If the demand function is:
Qd = 1300 - 10p - p^2
1. Sketch the demand curve above.
2. Find the arc elasticity of demand for the price range 8 to 12.
To sketch the demand curve, we need to plot the quantity (Q) on the y-axis and the price (p) on the x-axis. We can start by setting up a table of values and then plotting them on a graph.
Let's find some points on the demand curve:
- When p = 0, Qd = 1300 - 10(0) - 0^2 = 1300. So, one point is (0, 1300).
- When p = 10, Qd = 1300 - 10(10) - 10^2 = 200. So, another point is (10, 200).
- When p = 20, Qd = 1300 - 10(20) - 20^2 = -400. So, another point is (20, -400).
Now, let's plot these points and draw a smooth curve connecting them.
```
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0 10 20
```
This graph illustrates the demand curve.
To find the arc elasticity of demand for the price range 8 to 12, we can use the formula for arc elasticity of demand:
Arc elasticity of demand = (ΔQ/Δp) * (p/Q)
Where ΔQ is the change in quantity demanded and Δp is the change in price. Let's calculate it:
ΔQ = Q2 - Q1 = (1300 - 10(12) - 12^2) - (1300 - 10(8) - 8^2)
= -524 - 104 - 144 + 348 - 80 - 64
= -228
Δp = p2 - p1 = 12 - 8 = 4
Arc elasticity of demand = (-228/4) * (8/((1300 - 10(8) - 8^2)))
= -57 * (8/348)
= -1.31
The arc elasticity of demand for the price range 8 to 12 is approximately -1.31.