The area of the wall you are going to paint is 50 feet2. If the width is half the height, what are the dimensions of the wall? In answering this problem follow the steps below:

1. Represent the height as ‘x’ and then write the width in terms of x (for example, 2x, 5.5x, 0.5x, 0.3x).
2. Write down the formula for a rectangle’s area.
3. Substitute the values from step 1 and the original problem into the area formula.

What kind of equation do you need to solve in the end of step 3? How would you go about solving that equation? What other kinds of problems would lead you to this type of equation? Finally, what are the answers for the width and height?

Can you please explain step by step how you got the results? So I can best understand how to solve a similar problem? Thanks!

I will follow the suggested steps even though I don't think they are the best approach.

(I will do it the better way after)

let the height be x
then the width = (1/2)x (it said it was half the height)

(1/2)(x)(x) = 50
times 2
x^2 = 100
x=√100 = 10

so the height is 10
and the width is (1/2)(10) = 5

other way: (notice no fractions)
let the width be x
then the height is 2x (notice the width is 1/2 the height)
2x^2 = 50
x^2 = 25
x = √25 = 5

so the width is 5 and the
height is 2x = 10

Sure! I'll break down each step for you:

1. Represent the height as ‘x’ and then write the width in terms of x. Let's say the height of the wall is 'x'. According to the problem, the width is half the height, so the width can be represented as 0.5x.

2. Write down the formula for a rectangle's area. The formula for the area of a rectangle is length multiplied by width. In this case, it would be x * 0.5x.

3. Substitute the values from step 1 and the original problem into the area formula. The problem mentions that the area of the wall is 50 ft^2. So we can set up the equation as follows:

x * 0.5x = 50

Now, we need to determine what type of equation we have and how to solve it.

This equation is a quadratic equation because it contains the variable 'x' squared (0.5x * x). We can rearrange the equation into standard quadratic form:

0.5x^2 = 50

To solve this type of equation, we can multiply both sides by 2 to eliminate the fraction:

2 * 0.5x^2 = 2 * 50
x^2 = 100

Now, take the square root of both sides to solve for 'x':

sqrt(x^2) = sqrt(100)
x = 10

Since we know that the width is half the height, the width would be 0.5 times the height:

Width = 0.5 * 10 = 5 feet

Therefore, the dimensions of the wall are 10 feet (height) and 5 feet (width).

To summarize:
- Represent the height as 'x' and the width as 0.5x.
- Write down the formula for the area of a rectangle.
- Substitute the values into the area formula to set up the equation.
- Rearrange the equation into standard quadratic form and solve using appropriate techniques (in this case, squaring and square rooting).
- Determine the values for width and height using the solution obtained.

This type of equation can arise in various geometric or algebraic problems involving rectangles or quadratic equations. The specific technique used to solve will depend on the form of the equation.

Sure! Let's go step by step to solve the problem.

Step 1: Represent the height as 'x' and write the width in terms of 'x':
Given that the width is half the height, we can represent the width as 0.5x.

Step 2: Write down the formula for a rectangle's area:
The formula for the area of a rectangle is length multiplied by width. In this case, the length corresponds to the height and the width corresponds to 0.5 times the height. Therefore, the formula for the area is: Area = x * (0.5x).

Step 3: Substitute the values from step 1 and the original problem into the area formula:
We know that the area of the wall is 50 square feet. So, we can set up the equation as follows: 50 = x * (0.5x).

Now, let's analyze the equation we need to solve in step 3. It is a quadratic equation since we have an x term and an x^2 term. In this case, we have a quadratic equation that can be simplified by multiplying through by 2 to remove the fraction.

50 = x * (0.5x) becomes 50 = 0.5x^2.

To solve this quadratic equation, we can rearrange the equation in the standard form, which is ax^2 + bx + c = 0. In this case, we have 0.5x^2 - 50 = 0. Now, we can use various methods like factoring, completing the square, or using the quadratic formula to find the solutions. Let's use the quadratic formula.

The quadratic formula is x = (-b ± √(b^2 - 4ac)) / (2a).
In our case, a = 0.5, b = 0, and c = -50.

Substituting the values into the quadratic formula, we have:
x = (-0 ± √(0^2 - 4 * 0.5 * -50)) / (2 * 0.5).

Now, we simplify the equation:
x = ± √(0 - (-100)) / 1.

Simplifying further:
x = ± √100 / 1,
x = ±10 / 1,
x = ±10.

So, the possible values for the height 'x' are 10 and -10.

Finally, let's find the dimensions of the wall. The width is defined as half the height, so if the height is 10, the width is 0.5 * 10 = 5. Therefore, the dimensions of the wall are 10 feet (height) x 5 feet (width).

It is important to note that in this problem, the negative solution for 'x' is extraneous. In the context of finding dimensions, we consider only positive values since dimensions cannot be negative.

I hope this step-by-step explanation helps you understand how to solve similar problems! If you have any further questions, feel free to ask!