Sure! To determine after how many weeks the sunflowers in Class A and Class B will have the same height, we can set up equations based on their respective growth rates and initial heights and solve for the time in weeks.
Let's start by defining the variables and the growth equations:
- Let [tex]\( h_A \)[/tex] be the height of Class A's sunflower.
- Let [tex]\( h_B \)[/tex] be the height of Class B's sunflower.
- Let [tex]\( t \)[/tex] be the number of weeks.
We have the following information:
- The initial height of Class A's sunflower: [tex]\( 8 \)[/tex] centimeters.
- The weekly growth rate of Class A's sunflower: [tex]\( 3 \frac{1}{2} = 3.5 \)[/tex] centimeters per week.
- The initial height of Class B's sunflower: [tex]\( 10 \)[/tex] centimeters.
- The weekly growth rate of Class B's sunflower: [tex]\( 3 \frac{1}{4} = 3.25 \)[/tex] centimeters per week.
We express the height of each sunflower as a function of time [tex]\( t \)[/tex] in weeks:
[tex]\[ h_A(t) = 8 + 3.5t \][/tex]
[tex]\[ h_B(t) = 10 + 3.25t \][/tex]
We want to find the time [tex]\( t \)[/tex] when the heights of the sunflowers are the same, so we set [tex]\( h_A(t) = h_B(t) \)[/tex]:
[tex]\[ 8 + 3.5t = 10 + 3.25t \][/tex]
Next, solve this equation for [tex]\( t \)[/tex]:
1. Subtract [tex]\( 3.25t \)[/tex] from both sides to isolate terms involving [tex]\( t \)[/tex] on one side:
[tex]\[ 8 + 3.5t - 3.25t = 10 \][/tex]
[tex]\[ 8 + 0.25t = 10 \][/tex]
2. Subtract 8 from both sides to isolate [tex]\( t \)[/tex]:
[tex]\[ 0.25t = 10 - 8 \][/tex]
[tex]\[ 0.25t = 2 \][/tex]
3. Divide both sides by [tex]\( 0.25 \)[/tex] to solve for [tex]\( t \)[/tex]:
[tex]\[ t = \frac{2}{0.25} \][/tex]
[tex]\[ t = 8 \][/tex]
So, after 8 weeks, the heights of the sunflowers in Class A and Class B will be the same. The final answer is:
[tex]\[ t = 8 \text{ weeks} \][/tex]
