An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (1)
To find the volume V ( x ) , multiply the height function H ( x ) and the base area function A ( x ) .
Perform the multiplication: V ( x ) = ( x + 6 ) × 3 20 x .
Distribute and simplify: V ( x ) = 3 20 x x + 3 120 x .
Simplify the expression to find the volume function: V ( x ) = 3 20 x x + 40 x .
V ( x ) = 3 20 x x + 40 x
Explanation
Understanding the Problem The problem states that the height of the chest is given by H ( x ) = x + 6 and the area of the base is given by A ( x ) = 3 20 x . We need to find the volume V ( x ) of the chest. Since the chest is rectangular, its volume is the product of its base area and height. Therefore, we need to multiply the two functions to find the volume.
Calculating the Volume To find the volume V ( x ) , we multiply the functions H ( x ) and A ( x ) :
V ( x ) = H ( x ) × A ( x ) = ( x + 6 ) × 3 20 x Now, we distribute the 3 20 x across the terms in the parenthesis: V ( x ) = x × 3 20 x + 6 × 3 20 x V ( x ) = 3 20 x x + 3 120 x V ( x ) = 3 20 x x + 40 x
Final Answer The volume function is V ( x ) = 3 20 x x + 40 x . Therefore, Ronald should multiply the functions H ( x ) and A ( x ) , and the function that represents the volume of the chest is V ( x ) = 3 20 x x + 40 x .
Examples
Imagine you're designing a garden bed and want to calculate the volume of soil you need. If the height of the bed is represented by H ( x ) = x + 2 feet and the base area is A ( x ) = 5 x square feet, where x is the width, then the total volume of soil needed is V ( x ) = H ( x ) × A ( x ) = ( x + 2 ) × 5 x = 5 x x + 10 x cubic feet. This calculation helps you determine the amount of soil to purchase, ensuring you have enough to fill the garden bed properly. Understanding how to combine functions in this way is useful in many practical design and planning scenarios.
