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A company has learned that by pricing a newly released toy at $6, sales will reach 2500 per day. Raising the price to $8 will cause the
sales to fall to 1500 per day. Assume that the ratio of change in price to change in daily sales is constant and let x be the price of the toy
and y be the number of sales.
a. Find the linear equation that models the price-sales relationship for this toy. (Hint: The line must pass through (6,2500) and (8,1500).]
b. Use this equation to predict the daily sales of the toy if the price is set at $6.50

Answer :

Using linear functions, we have that:

a) The equation for the line is: [tex]y = -500x + 5500[/tex]

b) The prediction is of 2250 sales with a price of $6.50.

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The equation of a line is given by:

[tex]y = mx + b[/tex]

In which

  • m is the slope, which is the rate of change.
  • b is the y-intercept, which is the value of y when x = 0.

Item a:

In this problem:

  • There are two points (6, 2500) and (8, 1500).
  • The slope is given by the change in y divided by the change in x, then:

[tex]m = \frac{1500 - 2500}{8 - 6} = -\frac{1000}{2} = -500[/tex]

Thus:

[tex]y = -500x + b[/tex]

Point (6,2500) means that when [tex]x = 6, y = 2500[/tex], which we use to find b.

[tex]y = -500x + b[/tex]

[tex]2500 = -500(6) + b[/tex]

[tex]b = 5500[/tex]

Thus

[tex]y = -500x + 5500[/tex]

Item b:

  • The sales is y when x = 6.5, thus:

[tex]y = -500(6.5) + 5500 = 2250[/tex]

The prediction is of 2250 sales with a price of $6.50.

A similar problem is given at https://brainly.com/question/21010520

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