Exercises 49-54: Write a formula for a linear function that...

Exercises $49-54:$ Write a formula for a linear function that models the situation. Choose both an appropriate name and an appropriate variable for the function. State what the input variable represents and the domain of the function. Assume that the domain is an interval of the real numbers. Draining a Water Tank A 300 -gallon tank is initially full of water and is being drained at a rate of 10 gallons per minute. (a) Write a formula for a function $W$ that gives the number of gallons of water in the tank after $t$ minutes. (b) How much water is in the tank after 7 minutes? (c) Graph $W$ and identify and interpret the intercepts. (d) Find the domain of $W$.

Exercises $49-54:$ Write a formula for a linear function that models the situation. Choose both an appropriate name and an appropriate variable for the function. State what the input variable represents and the domain of the function. Assume that the domain is an interval of the real numbers.
Draining a Water Tank A 300 -gallon tank is initially full of water and is being drained at a rate of 10 gallons per minute.
(a) Write a formula for a function $W$ that gives the number of gallons of water in the tank after $t$ minutes.
(b) How much water is in the tank after 7 minutes?
(c) Graph $W$ and identify and interpret the intercepts.
(d) Find the domain of $W$.

Key Concepts

Linear Functions

A linear function represents a relationship between two quantities with a constant rate of change and can be expressed in the form f(x) = mx + b, where m is the slope and b is the y-intercept. In modeling real-world situations, this type of function is useful for predicting outcomes based on a change in the input variable.

Rate of Change

The rate of change, or slope, in a linear function indicates how much the dependent variable (output) changes for each unit increase in the independent variable (input). It is a crucial concept because it directly relates to the speed, speed, or efficiency of change in the situation being modeled.

Intercepts

Intercepts are points where the graph of a function crosses the axes. The y-intercept gives the initial value of the quantity being modeled when the independent variable is zero, while the x-intercept indicates the value of the independent variable at which the dependent variable becomes zero. Proper interpretation of these points is key in understanding the starting conditions and the limits of the modeled situation.

Domain of a Function

The domain of a function is the set of all possible input values for which the function is defined. In real-world applications, such as modeling physical processes, the domain is often restricted based on practical constraints or initial conditions. This careful consideration ensures that the model only predicts outcomes in a realistic and applicable range.

Function Notation and Variable Interpretation

Using appropriate function notation and clearly defining the input variable and its meaning is essential in mathematical modeling. This involves naming the function to reflect the phenomenon being modeled, determining the variable that represents the independent quantity (e.g., time), and stating what that variable represents, which helps in accurately interpreting the results of the model.