Question

Albert has $105 to spend on new basketball shoes. On his favorite shoe website, the prices for a pair of the shoes range from $80 to $180. The delivery fee is one-twentieth of the price of the basketball shoes. The total cost can be modeled by the function, f(c), where c represents the price of the basketball shoes. The domain of the function is [ , ]. Before purchasing the shoes, Albert remembered that his friend had to repay him an amount of money of $42. After receiving this amount, the domain for the function to represent the price of the basketball shoes that Albert can afford is [ , ].

Answer 1

Scenario 1

It is given that on Albert's favorite shoe website, the prices for a pair of the shoes range from $80 to $180 and the delivery fee is one-twentieth of the price of the basketball shoes.

We know that Albert has $105 to spend on new basketball shoes.

From the above pieces of information we see that the minimum that Albert will have to spend is $80+\frac{1}{12}\times 80=80+6.67=86.67$ dollars.

Now, we know that Albert can spend a maximum of $105 including the delivery fee. Let the upper limit of the price of the shoe Albert can buy be $x$. So, the upper limit of the domain can be found as:

$x+\frac{1}{12}x=105$

$\frac{13}{12}x=105$

$\therefore x=\frac{105\times 12}{13}\approx 96.92$ dollars.

Thus, in the first scenario, the domain of the total cost function, f(c) will be [86.67,96.92].

Scenario 2

After receiving $42 from his friend, Albert's total buying power becomes $147. Albert can now buy a costlier pair of shoes.

Thus, the maximum that Albert can buy is again given by:

$x+\frac{1}{12}x=147$

Solving this we get: $x=135.69$ dollars

The lower limit will remain the same as the lowest price point in the website is $80. Therefore, in the second scenario the domain is:

[86.67, 135.69]