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?
8400
A) To find the linear function that relates the demand for the watch to its price, we can use the slope-intercept form of a linear equation, which is y = mx + b. In this case, the demand (y) is the dependent variable and the price (x) is the independent variable.
First, let's find the slope (m) using the given information. We have two points: (88, 2000) and (38, 12000).
The slope (m) can be calculated as:
m = (y2 - y1) / (x2 - x1)
m = (12000 - 2000) / (38 - 88)
m = 10000 / (-50)
m = -200
Now, let's substitute one of the points into the equation to find the y-intercept (b). Using the point (88, 2000), we have:
2000 = -200(88) + b
2000 = -17600 + b
b = 19600
Therefore, the linear function that gives the demand for the watch (y) in terms of its price (x) is:
y = -200x + 19600
B) To find the price (x) when the demand (y) is 8,000, we can rearrange the equation to solve for x:
8000 = -200x + 19600
Subtract 19600 from both sides:
8000 - 19600 = -200x
Simplify:
-11600 = -200x
Divide both sides by -200:
x = 58
Therefore, the price when the demand is 8,000 would be $58.
C) To find the demand when the price is $65, we substitute x = 65 into the linear function:
y = -200(65) + 19600
y = -13000 + 19600
y = 6600
Therefore, the demand would be 6,600 if the company priced the watch at $65.