Question

Bryan bought a box of biscuits for his dog, Anchor. Anchor always gets three biscuits a day. At the start of the tenth day after opening the box, Bryan counts 106 biscuits left in the box. Write an equation to represent the number of biscuits left in the box in slope intercept form of the time, in days.

define the domain and range for this function

Answer 1

Let's denote the number of biscuits left in the box as "y" and the number of days since opening the box as "x". We want to write an equation in the slope-intercept form of a linear equation, which is y = mx + b, where "m" represents the slope and "b" represents the y-intercept.

We know that Anchor gets three biscuits a day, so we can deduce that the number of biscuits left in the box decreases by 3 each day. Therefore, the slope of the linear equation is -3.

On the tenth day, Bryan counts 106 biscuits left in the box. So, when x = 10 (in days), y = 106. This gives us an ordered pair (10, 106).

Now, we can substitute the slope (-3) and the ordered pair (10, 106) into the slope-intercept form equation to find the y-intercept (b):

106 = -3(10) + b
106 = -30 + b
b = 136

Therefore, the equation that represents the number of biscuits left in the box as a function of time, in days, is:

y = -3x + 136

Domain: The domain of this function represents the range of possible values for "x", which is the number of days since opening the box. Since the number of days cannot be negative and there is no known limit to the number of days, the domain is all non-negative real numbers or [0, ∞).

Range: The range of this function represents the possible values for "y", which is the number of biscuits left in the box. Considering the context of the problem, the number of biscuits left cannot be negative and is limited by the initial number of biscuits in the box. Therefore, the range is all non-negative integers between 0 and the initial number of biscuits in the box.