A hot-dog stand owner makes a profit of $100 when he sells 90 hot dogs a day. He has a loss of $30 when he sells 25
hot dogs a day. Write a linear equation that represents his profit?
No idea what I am supposed to do.
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100=90d
-30=25d
let x = number of hotdogs sold
let s = sell price of hotdog
let c = fixed operating costs
p = profit
p = x s - c
substitute to find s and c
100 = 90 s - c
-30 = 25 s - c
adding equations ... 130 = 65 s ... 2 = s
100 = 90 * 2 - c ... 80 = c
p = 2 x - 80
you have two ordered pairs of the type (h, p)
namely (90,100) and (25, -30)
slope = (-30-100)/(25-90) = 2
p-100 = 2(h-90)
p = 2h - 80 , which is what Scott had
Well, it looks like the hot-dog stand owner is quite the entrepreneur. To find a linear equation that represents his profit (P) based on the number of hot dogs sold (d), we can use the two points of information provided.
Let's start by creating a slope (m) using the formula = (change in profit) / (change in hot dogs sold):
m = (100 - (-30)) / (90 - 25) = 130 / 65 = 2
Now, we can use the point-slope formula to find the equation:
P - 100 = 2(d - 90)
Simplifying it:
P - 100 = 2d - 180
Rearranging the equation to get it in slope-intercept form (y = mx + b), where P is the y-value:
P = 2d - 80
So, the linear equation representing the hot-dog stand owner's profit is P = 2d - 80.
To find the linear equation that represents his profit, we need to find the slope and y-intercept.
The given information tells us that when he sells 90 hot dogs a day, he makes a profit of $100. So we can use the point (90, 100) to find the slope.
Using the formula for slope (m), which is (change in y)/(change in x), we can calculate the slope:
m = (100 - (-30))/(90 - 25)
m = 130/65
m = 2
Now that we have the slope, we can use the point-slope form of a linear equation:
y - y1 = m(x - x1)
Using the point (90, 100), we substitute the values into the equation:
y - 100 = 2(x - 90)
Simplifying:
y - 100 = 2x - 180
y = 2x - 80
Therefore, the linear equation that represents his profit is y = 2x - 80, where y is the profit and x is the number of hot dogs sold.
To write a linear equation that represents his profit, we can use the concept of slope-intercept form: y = mx + b.
In this case, the independent variable x represents the number of hot dogs sold per day, and the dependent variable y represents the profit made.
Let's find the slope, m:
Using the given information, when he sells 90 hot dogs a day, he makes a profit of $100. So we have the point (90, 100). In slope-intercept form, the slope (m) is calculated as the change in y divided by the change in x:
m = (y2 - y1) / (x2 - x1)
m = (100 - 0) / (90 - 0)
m = 100 / 90
m = 10/9
Now let's find the y-intercept, b:
Using the given information, when he sells 25 hot dogs a day, he has a loss of $30. So we have the point (25, -30). The y-intercept (b) is the value of y when x = 0. We can use the point-slope form to find b:
y - y1 = m(x - x1)
y - (-30) = (10/9)(x - 25)
y + 30 = (10/9)(x - 25)
y + 30 = (10/9)x - (10/9)(25)
y + 30 = (10/9)x - (250/9)
To simplify the equation, let's move the 30 to the other side:
y = (10/9)x - (250/9) - 30
y = (10/9)x - (250/9) - (270/9)
y = (10/9)x - (520/9)
Therefore, the linear equation that represents his profit is y = (10/9)x - (520/9).