The functions below represent the amounts of water released from two reservoirs over [tex]$x$[/tex] weeks:
Reservoir A
[tex]$f(x)=x^2-7 x+5$[/tex]
Reservoir B
[tex]$g(x)=3 x^2-6 x+2$[/tex]
The function [tex]$h(x)=f(x)-g(x)$[/tex] represents the difference in the amounts of water released.
Determine which statements about [tex]$h(x)$[/tex] and about the reservoirs are true. Check all that apply.
[tex]$h( x )=-2 x ^2-13 x +6$[/tex]
[tex]$h( x )=-2 x ^2- x +3$[/tex]
Reservoir A releases less water than Reservoir B over 1 week.
Reservoir A releases the same amount of water as Reservoir B over 1 week.
Reservoir A releases more water than Reservoir B over 1 week.
Answer (1)
Calculate h ( x ) = f ( x ) − g ( x ) , which results in h ( x ) = − 2 x 2 − x + 3 .
Evaluate f ( 1 ) and g ( 1 ) to find the amount of water released from each reservoir after 1 week: f ( 1 ) = − 1 and g ( 1 ) = − 1 .
Compare f ( 1 ) and g ( 1 ) to determine that Reservoir A and Reservoir B release the same amount of water after 1 week.
Determine the true statements based on the calculations: h ( x ) = − 2 x 2 − x + 3 and Reservoir A releases the same amount of water as Reservoir B over 1 week. h ( x ) = − 2 x 2 − x + 3 and Reservoir A releases the same amount of water as Reservoir B over 1 week.
Explanation
Understanding the Problem We are given two functions, f ( x ) = x 2 − 7 x + 5 and g ( x ) = 3 x 2 − 6 x + 2 , which represent the amount of water released from Reservoir A and Reservoir B, respectively, over x weeks. We are also given that h ( x ) = f ( x ) − g ( x ) represents the difference in the amounts of water released. We need to determine which statements about h ( x ) and the reservoirs are true.
Calculating h(x) First, let's find the expression for h ( x ) : h ( x ) = f ( x ) − g ( x ) = ( x 2 − 7 x + 5 ) − ( 3 x 2 − 6 x + 2 ) h ( x ) = x 2 − 7 x + 5 − 3 x 2 + 6 x − 2 h ( x ) = ( x 2 − 3 x 2 ) + ( − 7 x + 6 x ) + ( 5 − 2 ) h ( x ) = − 2 x 2 − x + 3
Evaluating f(1) and g(1) Now, let's evaluate f ( 1 ) and g ( 1 ) to determine the amount of water released from each reservoir after 1 week: f ( 1 ) = ( 1 ) 2 − 7 ( 1 ) + 5 = 1 − 7 + 5 = − 1 g ( 1 ) = 3 ( 1 ) 2 − 6 ( 1 ) + 2 = 3 − 6 + 2 = − 1
Comparing f(1) and g(1) Since f ( 1 ) = − 1 and g ( 1 ) = − 1 , Reservoir A and Reservoir B release the same amount of water after 1 week. Note that the negative value indicates that the water level is decreasing, but the amount released is the same.
Checking the Statements Now, let's check the given statements about h ( x ) :
h ( x ) = − 2 x 2 − 13 x + 6 is false, since we found h ( x ) = − 2 x 2 − x + 3 .
h ( x ) = − 2 x 2 − x + 3 is true, since we calculated h ( x ) = − 2 x 2 − x + 3 .
Reservoir A releases less water than Reservoir B over 1 week is false, since f ( 1 ) = g ( 1 ) .
Reservoir A releases the same amount of water as Reservoir B over 1 week is true, since f ( 1 ) = g ( 1 ) = − 1 .
Reservoir A releases more water than Reservoir B over 1 week is false, since f ( 1 ) = g ( 1 ) .
Final Answer Therefore, the true statements are:
h ( x ) = − 2 x 2 − x + 3
Reservoir A releases the same amount of water as Reservoir B over 1 week.
Examples
Understanding the difference between two functions, like the water release from two reservoirs, can help in managing resources effectively. For example, if you know how much water each reservoir releases over time, you can predict when one might need to be refilled or when there might be a surplus of water. This kind of analysis is used in many real-world situations, such as managing inventory, tracking sales, or even predicting population growth. By understanding the relationship between different factors, you can make better decisions and plan for the future. In this case, f ( x ) and g ( x ) represent the water released from two reservoirs, and by comparing these functions, we can determine which reservoir releases more water at a given time.
