Market research tells you that if you set the price of an item @ 1.50, you will be able to sell 5000 items; and for every 10 cents you lower the price below 1.50 you will be able to sell another 1000 items. Let x be the # of items you can sell, and P = the price of an item. A) Express P linearly in terms of x (in other words, in the form P = mx + b) B) express x linearly in terms of P. Thanks to anyone who can help!
To express P linearly in terms of x (P = mx + b), we need to determine the slope (m) and the y-intercept (b) from the given market research.
From the information provided, we know that if the price is $1.50, we can sell 5000 items. For every 10 cents decrease in price below $1.50, we can sell another 1000 items.
Let's use this information to calculate the slope (m):
For every 10-cent decrease in price, the number of items sold increases by 1000.
This means that the slope (m) can be calculated as the change in the number of items sold (Δx) divided by the corresponding change in price (ΔP).
Δx = 1000
ΔP = 0.10 (since a decrease in price is -0.10)
So, m = Δx / ΔP = 1000 / 0.10 = 10,000.
Now, to determine the y-intercept (b), we can substitute the values of P and x from one of the given data points. Taking the point (P, x) = ($1.50, 5000):
P = mx + b
$1.50 = 10,000 * 5000 + b
$1.50 = 50,000,000 + b
To isolate b, we subtract 50,000,000 from both sides:
b = $1.50 - 50,000,000
b ≈ -$49,999,998.50
Therefore, the linear equation expressing P in terms of x is:
P = 10,000x - $49,999,998.50
Now, to express x linearly in terms of P, we can rearrange the equation:
P = 10,000x - $49,999,998.50
Let's solve this equation for x:
x = (P + $49,999,998.50) / 10,000
Thus, the linear equation expressing x in terms of P is:
x = (P + $49,999,998.50) / 10,000