Converting Units There are 4 quarts in 1 gallon, 4 cups in 1 quart, and 16 tablespoons in 1 cup. (a) Write a function Q that converts x gallons to quarts.
(b) Write a function C that converts x quarts to cups.
(c) Write a function T that converts x cups to tablespoons.
(d) Express a function F that converts x gallons to tablespoons as a composition of three functions. (e) Write a formula for F.
The Correct Answer and Explanation is :
Let’s break down the problem step by step.
(a) Function Q that converts gallons to quarts:
We know that there are 4 quarts in 1 gallon. So, the function ( Q(x) ) that converts ( x ) gallons to quarts is:
[
Q(x) = 4x
]
(b) Function C that converts quarts to cups:
We know that there are 4 cups in 1 quart. So, the function ( C(x) ) that converts ( x ) quarts to cups is:
[
C(x) = 4x
]
(c) Function T that converts cups to tablespoons:
We know that there are 16 tablespoons in 1 cup. So, the function ( T(x) ) that converts ( x ) cups to tablespoons is:
[
T(x) = 16x
]
(d) Function F that converts gallons to tablespoons as a composition of three functions:
To express a function ( F ) that converts ( x ) gallons to tablespoons, we need to compose the functions we’ve created. First, we convert gallons to quarts, then quarts to cups, and finally cups to tablespoons.
The composition of functions will look like:
[
F(x) = T(C(Q(x)))
]
Where:
- ( Q(x) ) converts gallons to quarts,
- ( C(x) ) converts quarts to cups, and
- ( T(x) ) converts cups to tablespoons.
(e) Formula for F:
Now, let’s write out the full formula for ( F(x) ). We already know the formulas for ( Q(x) ), ( C(x) ), and ( T(x) ):
- ( Q(x) = 4x )
- ( C(x) = 4x )
- ( T(x) = 16x )
So, plugging in each function step by step:
[
F(x) = T(C(Q(x))) = T(C(4x)) = T(16x) = 16(16x) = 256x
]
Thus, the formula for ( F(x) ) is:
[
F(x) = 256x
]
Explanation:
In this problem, we are asked to convert units of volume. To do so, we use simple multiplication based on conversion factors:
- Gallons to quarts: There are 4 quarts in 1 gallon, so multiplying the number of gallons by 4 converts it to quarts.
- Quarts to cups: Similarly, there are 4 cups in 1 quart, so multiplying the number of quarts by 4 converts it to cups.
- Cups to tablespoons: Finally, there are 16 tablespoons in 1 cup, so multiplying the number of cups by 16 converts it to tablespoons.
For part (d), we used a composition of these functions to create a new function that converts directly from gallons to tablespoons. By composing the functions ( Q ), ( C ), and ( T ), we multiplied the factors together to get the final conversion factor, which in this case is 256 (since ( 4 \times 4 \times 16 = 256 )).
This approach shows how to convert between units by systematically applying conversion factors in sequence.