Question

A gardener is planting two types of trees: Type A is 8 feet tall and grows at a rate of 3 inches per year. Type B is 7 feet tall and grows at a rate of 4 inches per year. Algebraically determine exactly how many years it will take for these trees to be the same height.

Answer 1

Answer:

Exactly 12 years it will take for these trees to be the same height

Step-by-step explanation:

Slope intercept form: An equation of line is in the form of $y = mx+b$ where m is the slope or unit rate and b is the y-intercepts.

Let x represents the time in years and y represents the height of the tree.

Use conversion:

1 ft = 12 inches

As per the given statement:

Type A is 8 feet tall and grows at a rate of 3 inches per year.

⇒unit rate per year = 3 inches = $\frac{1}{4}$ ft

Then, we have;

$y =\frac{1}{4}x + 8$                      ......[1]

Similarly for;

Type B  is 7 feet tall and grows at a rate of 4 inches per year.

⇒unit rate per year = 4 inches = $\frac{1}{3}$ ft

then;

$y =\frac{1}{3}x + 7$                   .....[2]

To find after how many years it will take for these trees to be the same height.

Since, trees to be the same height;

⇒equate [1] and [2], to solve for x;

$\frac{1}{4}x + 8 = \frac{1}{3}x +7$

Subtract 7 from both sides we get;

$\frac{1}{4}x + 8-7= \frac{1}{3}x +7-7$

Simplify:

$\frac{1}{4}x + 1= \frac{1}{3}x$

Subtract $\frac{1}{4}x$ from both sides we get;

$1= \frac{1}{3}x-\frac{1}{4}x$

Simplify:

$1 = \frac{x}{12}$

Multiply both sides by 12 we get;

x = 12

Therefore, exactly it will take for these trees to be the same height is, 12 years