Math / Geometry

QuestionA rope is tied halfway to the top of an 8 m tree. The other end is tied to the ground 1 m from the tree. Which equation represents the rope's height $y$ at distance $x$ from the tree? A. $x-4 y=4$ B. $4 x-y=4$ C. $4 x+y=4$ D. $x-4 y=-4$

Studdy Solution

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 $4x - y = -4$ .

STEP 10

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

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QuestionA rope is tied halfway to the top of an 8 m tree. The other end is tied to the ground 1 m from the tree. Which equation represents the rope's height $y$ at distance $x$ from the tree? A. $x-4 y=4$ B. $4 x-y=4$ C. $4 x+y=4$ D. $x-4 y=-4$

Studdy Solution

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

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 $4x - y = -4$ .

STEP 10

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

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Studdy Solution

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 diagram2. Find the slope3. Find the y-intercept4. Construct the equation

STEP 3

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 41=4\frac{4}{1} = 414​=4.

STEP 5

The **y-intercept** is the height of the rope where it touches the tree (when the horizontal distance xxx 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+by = mx + by=mx+b, where mmm is the **slope** and bbb is the **y-intercept**.

STEP 7

We found that our **slope** mmm is **4** and our **y-intercept** bbb is also **4**.

STEP 8

Plugging these values into our equation gives us y=4x+4y = 4x + 4y=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 4x4x4x from both sides of the equation to get −4x+y=4-4x + y = 4−4x+y=4.Multiplying both sides by −1-1−1 gives us 4x−y=−44x - y = -44x−y=−4.

STEP 10

The equation that represents the rope's height is 4x−y=−44x - y = -44x−y=−4, which corresponds to answer choice **D**.

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Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

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

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 $\frac{4}{1} = 4$ .

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.

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.

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

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

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.