The price p (in dollars) and the quantity x sold of a certain product satisfy the demand equation x=−5p+200. Answer parts (a) through (g).
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What are "(a) through (g)"?
To solve the given problem, we need to answer parts (a) through (g) step-by-step. Here's how:
(a) Find the demand quantity for a price of $20.
To find the demand quantity, plug in the given price ($20) into the demand equation:
x = -5p + 200
x = -5(20) + 200
x = -100 + 200
x = 100
Therefore, the demand quantity for a price of $20 is 100.
(b) Find the price at which demand is zero.
To find the price at which demand is zero, set the demand quantity (x) equal to zero and solve for p:
x = -5p + 200
0 = -5p + 200
5p = 200
p = 200/5
p = 40
Therefore, the price at which demand is zero is $40.
(c) Find the slope of the demand equation.
The slope of the demand equation can be found by looking at the coefficient of the price (p) term in the equation. In this case, the coefficient is -5. Therefore, the slope of the demand equation is -5.
(d) Find the intercepts of the demand equation.
To find the intercepts, set x and p equal to zero separately and solve for the other variable:
When x = 0:
0 = -5p + 200
5p = 200
p = 200/5
p = 40
Therefore, the x-intercept is 40.
When p = 0:
x = -5(0) + 200
x = 200
Therefore, the y-intercept is 200.
(e) Express the demand equation in slope-intercept form.
The slope-intercept form of an equation is y = mx + b, where m is the slope and b is the y-intercept. In this case, x is equivalent to y, and p is equivalent to x. Therefore, the demand equation can be expressed as:
x = -5p + 200
y = -5x + 200
(f) Sketch the graph of the demand equation.
The demand equation y = -5x + 200 represents a linear equation in slope-intercept form, where the slope is negative and the y-intercept is positive. The graph will be a straight line that starts at the y-intercept (200) and decreases as x (or p) increases.
(g) Determine the price at which demand for the product is 150 units.
To determine the price at which demand for 150 units, plug in the given demand quantity (150) into the demand equation:
x = -5p + 200
150 = -5p + 200
-5p = 150 - 200
-5p = -50
p = -50/-5
p = 10
Therefore, the price at which demand for the product is 150 units is $10.