For each of the following situations, write down the formula and determine if it defines a linear function:
The length of a rectangle is 10 cm, while the width is x cm. The area of the rectangle is A. What is the formula for dependence of A on x?
recall that area is length times width.
Now write that in math.
To determine if the formula for the dependence of A (the area) on x (the width) defines a linear function, we first need to express A as a function of x.
In this case, the area of the rectangle is given by the formula: A = length × width.
From the given information, we know that the length is fixed at 10 cm, and the width is represented by x cm. So, the formula for the area becomes:
A = 10 cm × x cm
A = 10x cm²
Now, to determine if this defines a linear function, we need to check if the equation can be written in the form y = mx + b, where y represents the dependent variable (A), x represents the independent variable (x), m represents the slope, and b represents the y-intercept.
The equation A = 10x cm² can be rearranged as:
A = 10x^1 cm²
We can see that the exponent on x is 1, which means it is a linear term. Therefore, the formula A = 10x cm² defines a linear function.