Question

You are redesigning a rectangular raft. The raft is 6 feet long and 4 feet wide. You want to double the area of the raft by adding to the existing design. Draw a diagram of the new raft. Write and solve an equation you can use to find the dimensions of the new raft.

Short answer

The new dimensions of the raft are 12 feet in length and 4 feet in width.

Step-by-step solution

Calculate the Original Area

Find the area of the original raft using the formula \(Area = length \times width\). Plug in the given dimensions, \(Area = 6 feet \times 4 feet = 24 square feet\).

Double the Original Area

Since the new raft should have double the area of the original, calculate the new area by multiplying the original area by 2, \(New Area = 2 \times 24 square feet = 48 square feet\).

Formulate an Equation for the New Dimensions

Assume the length of the new raft to be \(x\) feet and the width remains the same (4 feet). Then the equation based on the area of the rectangle becomes \(48 square feet = x feet \times 4 feet\).

Solve for the New Dimensions

Solve the equation for \(x\) to find the new length. Divide both sides of the equation by 4 feet, \[x = \frac{48 square feet}{4 feet} = 12 feet\]. So, the length of the new raft will be 12 feet while the width remains the same, 4 feet.

Draw the New Raft

The exercise also asks for a diagram of the new raft. It should be a rectangle labeled with the new dimensions: 4 feet width and 12 feet length.

Key concepts

Area Calculation

Understanding how to calculate the area of a shape is fundamental in geometry, especially when dealing with rectangles. To find the area of a rectangle, you need to multiply its length by its width.
For our original raft, this calculation was simple. Given the length of 6 feet and width of 4 feet, the area was computed as follows:

• The formula to find the area is: \[ Area = ext{length} \times ext{width} \]
• Substituting the values, we found: \[ Area = 6 ext{ feet} \times 4 ext{ feet} = 24 ext{ square feet} \]

By understanding this basic arithmetic, we set the foundation for resizing the raft.

Rectangular Dimensions

Knowing how to work with the dimensions of a rectangle helps in understanding and solving various real-world geometry problems. The dimensions of a rectangle are its length and width. These two measurements define the size and shape of the rectangle.
In our exercise, the raft started with dimensions of 6 feet by 4 feet. When tasked with increasing the area, it's essential to understand that we can adjust these dimensions responsibly.
If one dimension is to change, the other can remain constant; hence, understanding which dimension to manipulate comes in handy.

• Length and width determine area.
• Adjust one to change the area without losing the rectangular shape.

Understanding these basics allows us to conceptualize potential changes and their implications.

Equation Formulation

Formulating equations is a fundamental step in solving geometry problems. Equations help us set up relationships between different dimensions and known variables.
For our problem, since the area needed to double to 48 square feet and the width remained constant at 4 feet, we introduced a new variable, \(x\), as the new length.
This gave us our equation:

• The new area of the rectangle: \(48 ext{ square feet} = x ext{ feet} \times 4 ext{ feet} \)

Doing this:

• Allows the problem to be reduced to a simple algebraic expression.
• Gives a means to solve for unknown dimensions.

Thus, formulating a correct equation paves the way for solving for the desired dimensions.

Problem-solving Steps

Solving geometry problems systematically involves a few key steps, beginning from understanding to execution.
Consider these structured steps from the exercise:

• Calculate the original area using known dimensions.
• Determine the new area requirement (in this case, doubling the original).
• Create a mathematical equation to link new dimensions to the desired area.
• Solve the equation to find unknown dimension(s).
• Apply the solution in a practical design or diagram.

Each step builds on the previous, ensuring a seamless transition from problem conception to solution execution.
Practicing this series of steps boosts problem-solving efficiency and accuracy.