Mrs. Fields bought a sapling from a tree farm nursery and observed a linear growth of the sapling over a period of 6 months. She found the height of the sapling, H centimeters, and the time, t months is related by the linear equation H=2(4+3t)
a) write an equation for t in terms of h
C) find h when t=6
D) how many months until the height of the sapling is 29 centimeters?
a) h = 8 + 6t
6t = h-8
t = (h-8)/6
c) when t = 6
h = 8 + 6(6) = ....
d) when h = 29
29 = 8 + 6t
21 = 6t
t = 21/6 = 3.5
it would take 3.5 months
a) To write an equation for t in terms of h, we need to isolate t on one side of the equation.
Given the equation H = 2(4 + 3t), let's solve it for t:
H = 2(4 + 3t)
Divide both sides by 2:
H/2 = 4 + 3t
Subtract 4 from both sides:
(H/2) - 4 = 3t
Divide both sides by 3:
[(H/2) - 4]/3 = t
So the equation for t in terms of h is:
t = [(H/2) - 4]/3
b) To find h when t = 6, we can substitute t = 6 into the equation H = 2(4 + 3t):
H = 2(4 + 3(6))
Simplify within the brackets:
H = 2(4 + 18)
H = 2(22)
H = 44
So when t = 6, h (the height of the sapling) is 44 centimeters.
c) To find how many months until the height of the sapling is 29 centimeters, we can use the equation H = 2(4 + 3t) and substitute H = 29:
29 = 2(4 + 3t)
Divide both sides by 2:
29/2 = 4 + 3t
Subtract 4 from both sides:
29/2 - 4 = 3t
Simplify:
29/2 - 8/2 = 3t
Combine the fractions:
(29 - 8)/2 = 3t
21/2 = 3t
Divide both sides by 3:
(21/2)/3 = t
To simplify (21/2)/3, we can multiply the numerator and denominator by 2 to get a common denominator:
(21/2)(2/2)/3 = t
(21 * 2)/(2 * 3) = t
(42/6) = t
7/1 = t
t = 7
So it will take 7 months until the height of the sapling is 29 centimeters.