Let's go through each part of the problem step-by-step.

Part (a): Expression for the Perimeter

The perimeter $P$ of a rectangle is given by the formula: $P = 2 \times (\text{length} + \text{height})$

Given:

• Length $= x + 7$
• Height $= x$

Substitute these values into the formula for the perimeter: $P = 2 \times ((x + 7) + x)$

Simplify the expression inside the parentheses: $P = 2 \times (2x + 7)$

Now distribute the 2: $P = 4x + 14$

So, the simplified expression for the perimeter of the rectangle is: $P = 4x + 14$

Part (b): New Perimeter When Length is Tripled

Now, we are told that the length $x + 7$ is tripled. The new length becomes: $3 \times (x + 7) = 3x + 21$

The height remains the same, $x$.

Substitute these new values into the perimeter formula: $P_{\text{new}} = 2 \times (\text{new length} + \text{height})$ $P_{\text{new}} = 2 \times ((3x + 21) + x)$

Simplify the expression inside the parentheses: $P_{\text{new}} = 2 \times (4x + 21)$

Now distribute the 2: $P_{\text{new}} = 8x + 42$

So, the simplified expression for the new perimeter of the rectangle is: $P_{\text{new}} = 8x + 42$

Would you like further details or have any questions?

5 Related Questions

1. What would be the area of the original rectangle in terms of $x$?
2. If the height of the rectangle was doubled instead of the length being tripled, what would the new perimeter be?
3. For what value of $x$ would the original perimeter be 50 units?
4. How does the change in perimeter affect the shape of the rectangle when only the length is altered?
5. What would the dimensions of the rectangle be if $x = 5$?

Tip

When dealing with algebraic expressions involving geometric figures, always start by writing down the formulas and substituting the given values carefully.