Math Snap

STEP 1

What is this asking?
We need to find the equation that describes how high up the rope is, given how far along the ground it is from the tree.
Watch out!
The problem says "halfway to the top", so be careful to use the correct tree height in your calculations!

STEP 2

1. Draw a diagram
2. Find the slope
3. Find the y-intercept
4. Construct the equation

STEP 3

Let's visualize this!
Imagine the tree, the rope, and the ground forming a right triangle.
The tree is 8 m tall, but the rope is tied halfway up, so it's attached at a height of $8 \div 2 = 4$ m.
The rope is tied to the ground 1 m from the tree.
Draw a right triangle with a vertical side of 4 m and a horizontal side of 1 m.

STEP 4

The slope tells us how steep the rope is.
It's the change in height divided by the change in horizontal distance.
In our triangle, the height changes by 4 m and the horizontal distance changes by 1 m.
So, the slope is $\frac{4}{1} = 4$.

STEP 5

The y-intercept is the height of the rope where it touches the tree (when the horizontal distance $x$ is zero).
Since the rope is attached to the tree at 4 m, our y-intercept is 4.

STEP 6

We can use the slope-intercept form of a linear equation: $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

STEP 7

We found that our slope $m$ is 4 and our y-intercept $b$ is also 4.

STEP 8

Plugging these values into our equation gives us $y = 4x + 4$.

STEP 9

The question wants the equation in a slightly different form.
Let's rearrange our equation to match the answer choices.
We can subtract $4x$ from both sides of the equation to get $-4x + y = 4$.
Multiplying both sides by $-1$ gives us $4x - y = -4$.

The equation that represents the rope's height is $4x - y = -4$, which corresponds to answer choice D.