Hailey’s new puppy, Sparky, weighed 4 pounds at six weeks old. If the puppy gains 2.5 pounds each month, at how many months will Sparky weigh 14 pounds and is it modeled by a linear function or a exponential function
Assuming a month is 4 weeks, after x weeks, Sparky weighs
4 + (5/2)(x/4)
So, when do you have
4 + 5/8 x = 14
That is the number of weeks, not months. Clearly it is linear, not exponential. (2.5 lbs/month)
2.5 pounds/month is a constant slope of weight as a function of age.
Therefore we are dealing with a straight line, linear function.
(eventually it will hopefully turn exponential as it approaches full grown weight but that is a later problem :)
six weeks = 1.5 months
weight = weight at 1.5 months plus slope *time after 1.5 months
or
w = 4 + 2.5 (t - 1.5)
w = 2.5 t + 4 - 3.75
w = 2.5 t + .25 in slope intercept form
14 = 2.5 t + .25
2.5 t = 13.75
t = 5.5 months
To determine at how many months Sparky will weigh 14 pounds and whether it is modeled by a linear or exponential function, we need to analyze the information given.
First, let's calculate the number of months it will take for Sparky to gain 10 pounds (from 4 pounds to 14 pounds), considering a monthly gain of 2.5 pounds.
10 pounds / 2.5 pounds per month = 4 months
Therefore, Sparky will take 4 months to weigh 14 pounds.
Now, let's determine whether this relationship is modeled by a linear or exponential function.
A linear function has a constant rate of change, meaning that for each month, Sparky would gain the same amount of weight. In this case, that would be 2.5 pounds per month.
An exponential function, on the other hand, has a variable rate of change, where the change is proportional to the current value. In this case, if the weight gain is described as exponential, it means that as Sparky gets bigger, the amount of weight gained each month increases.
Since Sparky gains a constant amount of weight each month (2.5 pounds), it is modeled by a linear function rather than an exponential function. The weight gain is consistent and not proportional to Sparky's current weight.
In summary, Sparky will weigh 14 pounds at 4 months old, and his weight gain is modeled by a linear function.