How do I solve the following word problem?
A botanist watched the growth of a lily. At 3 weeks, the lily was 4 inches tall. Four weeks later, the lily was 21 inches tall. Assuming the relationship is linear:
(a) Write an equation to show the grow pattern of this plant.
(b) How tall was the lily at the 5.5 mark?
(c) Is there a restriction on how high the plant will grow? Does your equation show this?
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First, assign the variables.
x: time of growth (weeks)
y: height of lily (inches)
Next, use the given information to write two points.
At 3 weeks, the lily was 4 inches tall. (3, 4)
Four weeks later, the lily was 21 inches tall. (7, 21)
Now use these points to calculate the slope of the line.
m = (y2 - y1) / (x2 - x1)
m = (21 - 4) / (7 - 3)
m = 17/4 = 4.25
Now substitute 4.25 in for m in the equation. Also, select one of the points and substitute 3 in for x and 4 in for y. Solve for b.
y = mx + b
4 = 4.25 * 3 + b
4 = 12.75 + b
-8.75 = b
The equation that represents the growth pattern of this plant is...
(a) y = 4.25x + -8.75
To find the height of the lily after 5.5 weeks, substitute 5.5 in for x and solve for y.
y = 4.25 * 5.5 + -8.75
y = 14.625
At the 5.5 mark, the height of the lily will be...
(b) 14.625 inches
There is a restriction on how high the plant will grow. A plant will not grow to an unlimited height. For example, after 75 weeks, the lily would reach a height of 310 inches. That's over 25 feet tall.
The equation does not show these restrictions. By definition, a linear equation continues indefinitely in both directions. Lilies grow to a maximum height of only a few feet.
(c) Yes, there is a restriction. No, the equation does not reflect this restriction.
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