1) a company manufactures and sells a specialty watch. the financial reserach department detemines that at the price of $88 each, the demand would be 2,000 watches and at $38 each the demand will be 12,000 watches.
A) assuming a linear relationship between price and demand, find a linear function which gives the demand for the watch in terms of its price.
B) What would be the price when the demand is 8,000?
C) What would be the demand, if the company priced the watch at $65?
Price=m*demand+b
put in both data points...
88=m2000+b
38=m12000+b
subtract the second equation from the first.
50=-10,000m
solve for m
then put either point back in the original equation, solve for b.
To find the linear function that gives the demand for the watch in terms of its price, we can use the formula for a linear equation: y = mx + b, where y represents the demand, x represents the price, m represents the slope, and b represents the y-intercept.
Step 1: Find the slope (m)
To find the slope, we can use the formula: m = (y2 - y1) / (x2 - x1), where (x1, y1) = ($88, 2000) and (x2, y2) = ($38, 12000).
m = (12000 - 2000) / (38 - 88)
m = 10000 / (-50)
m = -200
Step 2: Find the y-intercept (b)
We can now substitute the values of x and y from one of the given points into the equation to solve for b. Let's use the point (x, y) = ($38, 12000).
12000 = -200 * 38 + b
12000 = -7600 + b
b = 12000 + 7600
b = 19600
Therefore, the linear function that gives the demand in terms of the price is: y = -200x + 19600.
Now let's move on to the specific questions.
B) To find the price when the demand is 8,000, we can substitute y = 8000 into the equation and solve for x.
8000 = -200x + 19600
-200x = -8000 + 19600
-200x = 11600
x = 11600 / -200
x = -58
Therefore, the price when the demand is 8,000 would be $-58. However, negative prices do not make sense in this context, so we can conclude that there is no viable price when the demand is exactly 8,000.
C) To find the demand when the price is $65, we can substitute x = 65 into the equation and solve for y.
y = -200(65) + 19600
y = -13000 + 19600
y = 6600
Therefore, if the company priced the watch at $65, the demand would be 6,600 watches.