To model the height of the bean plant, [tex]\( y \)[/tex], after [tex]\( x \)[/tex] days, we need to find the equation of the line that passes through the given points. Here's a detailed step-by-step solution:
1. Identify the given points:
We have two points representing the height of the plant at different times:
- After 10 days, the height [tex]\( y \)[/tex] is 25 cm: [tex]\((x_1, y_1) = (10, 25)\)[/tex]
- After 20 days, the height [tex]\( y \)[/tex] is 45 cm: [tex]\((x_2, y_2) = (20, 45)\)[/tex]
2. Calculate the slope (rate of growth):
The slope, [tex]\( m \)[/tex], of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting the given points:
[tex]\[
m = \frac{45 - 25}{20 - 10} = \frac{20}{10} = 2
\][/tex]
So, the slope of the line is [tex]\( m = 2 \)[/tex].
3. Use the point-slope form to find the equation:
The point-slope form of the equation of a line is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Substitute [tex]\( m = 2 \)[/tex], [tex]\( x_1 = 10 \)[/tex], and [tex]\( y_1 = 25 \)[/tex] into the equation:
[tex]\[
y - 25 = 2(x - 10)
\][/tex]
4. Choose the correct option:
Looking at the given options, the equation we derived matches option B:
[tex]\[
y - 25 = 2(x - 10)
\][/tex]
Therefore, the correct equation that models the height of the plant, [tex]\( y \)[/tex], after [tex]\( x \)[/tex] days is:
[tex]\[
\boxed{y - 25 = 2(x - 10)}
\][/tex]
This corresponds to option B.
