Question

A large apple tree may absorb 360 liters of water from the soil per day. The amount of water W absorbed over a short period of time is modeled by the function W = 360d, where d represents the number of days. Copy and complete the table. $$ \begin{array}{|l|l|l|} \hline \text { Input } & \text { Function } & \text { Output } \\ \hline d=1 & W=360 \cdot 1 & W=360 \\ \hline d=2 & W=? & W=? \\ \hline d=3 & W=? & W=? \\ \hline d=4 & W=? & W=? \\ \hline d=5 & W=? & W=? \\ \hline \end{array} $$

Short answer

The completed table is\[\begin{array}{|l|l|l|}\hline \text{Input} & \text{Function} & \text{Output} \\hline d=1 & W=360 \cdot 1 & W=360 \\hline d=2 & W=360 \cdot 2 & W=720 \\hline d=3 & W=360 \cdot 3 & W=1080 \\hline d=4 & W=360 \cdot 4 & W=1440 \\hline d=5 & W=360 \cdot 5 & W=1800 \\hline \end{array}\]

Step-by-step solution

- Understand the Function

The function W = 360d tells us that for every day (d), 360 liters of water are absorbed by the tree. This function should be used to find the amount of water (W) absorbed for various number of days.

- Apply the Function for d=2

To find the water absorbed when d=2, substitute d=2 into the function: W = 360*(2) = 720 liters.

- Apply the Function for d=3

To calculate the amount of water absorbed when d=3, substitute d=3 into the function: W = 360*(3) = 1080 liters.

- Apply the Function for d=4

To find the water absorbed when d=4, substitute d=4 into the function: W = 360*(4) = 1440 liters.

- Apply the Function for d=5

To find the amount of water absorbed when d=5, put d=5 in the function: W = 360*(5) = 1800 liters.

Key concepts

Algebraic Functions

Understanding algebraic functions is crucial in mathematics, and it is especially pertinent when we consider problems like the apple tree water absorption model. An algebraic function is an expression that relates an input to an output using algebraic operations, which typically include adding, subtracting, multiplying, and dividing, as well as raising to powers.

An essential characteristic of algebraic functions is their use of variables, which are symbols that represent unknown values. In the case of our apple tree, the variable 'd' represents the number of days, which is our input. The function itself, defined by the equation W = 360d, uses multiplication to relate 'd' to 'W', the output representing the amount of water absorbed in liters.

These functions can be simple or complex, but what they all have in common is the ability to provide a clear mathematical relationship between quantities. And so, they allow us to model real-world scenarios predictably and efficiently.

Function Application

To utilize a mathematical function, one must engage in function application, a process that involves assigning a value to the input variable and calculating the corresponding output. This process turns abstract expressions into concrete numbers that describe real situations.

For our apple tree scenario, the function application involves substituting a specific number of days for 'd' in the equation W = 360d. For example, for d=2, we replace 'd' with 2, yielding the calculation W = 360*2, which simplifies to 720 liters. This transformation from algebraic expression to numerical result is at the core of using functions to solve problems.

It's essential to properly perform substitution and carry out the algebraic operations while applying functions. Doing this correctly ensures accurate predictions and solutions within the context of the model being used.

Linear Models

Linear models are a type of algebraic function that has a constant rate of change and can be represented with a straight line when graphed. They are given by the general formula y = mx + b, where 'm' is the slope of the line, and 'b' is the y-intercept. In linear models, the slope is the rate of change, indicating how the output (y) changes for each unit increase in the input (x).

In the exercise, the function W = 360d is a linear model. It tells us that the water absorbed by the tree (W) increases by a steady amount of 360 liters per day, with the number of days (d) being our input variable. There is no y-intercept (b) here because the function passes through the origin, assuming that no water is absorbed when there are zero days.

Linear models like this are invaluable in understanding relationships where the change between variables is steady and predictable, making them ideal for planning, forecasting, and analyzing patterns in various fields such as economics, biology, and physics.