Select the correct answer. Justin and Kira use functions to model the heights, in centimeters, of two sunflower plants [tex]$x$[/tex] weeks after transplanting them to the school garden. Function [tex]$J$[/tex] models the height of Justin's plant: [tex]$J(x)=18+6 x$[/tex] Function [tex]$k$[/tex] models the height of Kira's plant: [tex]$k(x)=12+4 x$[/tex] Which function correctly represents how much taller Justin's plant is than Kira's plant, x weeks after they were transplanted to the school garden? A. [tex]$(J-k)(x)=30+10 x$[/tex] B. [tex]$(J-k)(x)=6+10 x$[/tex] C. [tex]$(J-k)(x)=30+2 x$[/tex] D. [tex]$(J-k)(x)=6+2 x$[/tex]
Answer (2)
Subtract Kira's plant height function from Justin's plant height function: ( J − K ) ( x ) = J ( x ) − K ( x ) .
Substitute the given functions: ( J − K ) ( x ) = ( 18 + 6 x ) − ( 12 + 4 x ) .
Simplify the expression by combining like terms: ( J − K ) ( x ) = 6 + 2 x .
The correct function representing the height difference is ( J − K ) ( x ) = 6 + 2 x .
Explanation
Understanding the Problem We are given two functions, J ( x ) = 18 + 6 x representing the height of Justin's plant and K ( x ) = 12 + 4 x representing the height of Kira's plant. We want to find the function that represents the difference in height between Justin's plant and Kira's plant, which is J ( x ) − K ( x ) .
Setting up the Subtraction To find the difference in height, we subtract Kira's plant height function from Justin's plant height function: ( J − K ) ( x ) = J ( x ) − K ( x ) Substituting the given functions, we get: ( J − K ) ( x ) = ( 18 + 6 x ) − ( 12 + 4 x ) Now, we simplify the expression by combining like terms.
Simplifying the Expression ( J − K ) ( x ) = 18 + 6 x − 12 − 4 x ( J − K ) ( x ) = ( 18 − 12 ) + ( 6 x − 4 x ) ( J − K ) ( x ) = 6 + 2 x
Finding the Correct Option The function that represents how much taller Justin's plant is than Kira's plant is ( J − K ) ( x ) = 6 + 2 x . Comparing this to the given options, we see that option D matches our result.
Examples
Understanding the difference between two linear functions, like the heights of plants over time, can help in various real-world scenarios. For example, if you're tracking the growth of two investments, where each grows linearly, this calculation helps determine how much one investment outperforms the other over time. Similarly, in manufacturing, if two machines produce items at different linear rates, this calculation can show the difference in their output after a certain period. This concept is also applicable in comparing distances covered by two objects moving at different constant speeds.
After performing the calculation, we found that the height difference between Justin's plant and Kira's plant is represented by the function ( J − K ) ( x ) = 6 + 2 x . The correct answer is option D. This means that Justin's plant is always 6 centimeters taller than Kira's plant, increasing by 2 centimeters each week.
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