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Choose a value for a that is greater than 5 and less than 15 ( use 10 ) . Evaluate the

perimeter of the pen for that value, using one of the expressions that you
wrote. (use a + b + a + b = 2a + 2b)

11 answers

To evaluate the perimeter of the pen using the expression \( P = 2a + 2b \), we need both the length \( a \) and the width \( b \) of the pen.

Given \( a = 10 \), we still need a value for \( b \). Let's choose a value for \( b \) that is also reasonable for a pen; for example, we can say \( b = 5 \).

Now, we can calculate the perimeter \( P \):

\[ P = 2a + 2b \]

Substituting the values of \( a \) and \( b \):

\[ P = 2(10) + 2(5) \] \[ P = 20 + 10 \] \[ P = 30 \]

Thus, the perimeter of the pen with \( a = 10 \) and \( b = 5 \) is \( 30 \) units.

also here is the past question if it helps

A’Leila is building a pen for her pet donkey. The pen is a rectangle with one side
measuring b yards and the adjacent side measuring a yards. A’Leila knows that a = 1/3b

Thanks for providing the additional context regarding the relationship between the sides of the rectangle. We know that:

  • \( a = \frac{1}{3}b \)

To find the perimeter using this relationship, we can express the perimeter in terms of \( b \):

  1. Start with the formula for perimeter: \[ P = 2a + 2b \]

  2. Substitute \( a = \frac{1}{3}b \) into the perimeter formula: \[ P = 2\left(\frac{1}{3}b\right) + 2b \]

  3. Simplify the equation: \[ P = \frac{2}{3}b + 2b \] To add these, convert \( 2b \) into a fraction with a denominator of 3: \[ P = \frac{2}{3}b + \frac{6}{3}b = \frac{8}{3}b \]

Now, let's evaluate the perimeter using \( b \) when \( a = 10 \) yards. From the relationship \( a = \frac{1}{3}b \), we can find \( b \):

\[ 10 = \frac{1}{3}b \implies b = 10 \times 3 = 30 \text{ yards} \]

Now we calculate the perimeter \( P \) using \( b = 30 \):

\[ P = \frac{8}{3}(30) \] \[ P = \frac{240}{3} = 80 \text{ yards} \]

Thus, the perimeter of the pen, given that \( a = 10 \) and using the relationship \( a = \frac{1}{3}b \), is 80 yards.

get rid of the ( { [ so its easier for me to read

Sure! Let’s rewrite the calculations without parentheses for better readability.

Given:

  • a = 1/3b

Perimeter Formula:

The perimeter, P, of the rectangle can be expressed as: P = 2a + 2b

Substitute a in terms of b:

P = 2 * (1/3)b + 2b

Simplifying the equation:

P = (2/3)b + 2b

Convert 2b to a fraction with a denominator of 3: P = (2/3)b + (6/3)b

Now combine the fractions: P = (8/3)b

Now, find b when a = 10:

From the relationship a = 1/3b: 10 = (1/3)b

To find b, multiply both sides by 3: b = 10 * 3 b = 30

Calculate the perimeter using b = 30:

P = (8/3)(30)

Now simplify: P = 240/3 P = 80

Conclusion:

The perimeter of the pen is 80 yards.

your using the " a+ b + a + b = 2a + 2b " right?

Yes, that's correct! The formula \( P = a + b + a + b \) simplifies to \( P = 2a + 2b \), which is used to calculate the perimeter of a rectangle where \( a \) is one side and \( b \) is the adjacent side.

Let's clarify and follow this overall process step by step using the relationship \( a = \frac{1}{3}b \) and the perimeter calculation.

  1. Given the relationship:

    • \( a = \frac{1}{3}b \)

  2. Perimeter formula: Using the formula for perimeter: \[ P = a + b + a + b = 2a + 2b \]

  3. Substituting \( a \): Substitute \( a \) in terms of \( b \):

    • Since \( a = \frac{1}{3}b \), we substitute \( a \) in the perimeter formula: \[ P = 2\left(\frac{1}{3}b\right) + 2b \]

  4. Simplifying the formula: \[ P = \frac{2}{3}b + 2b \] Now convert \( 2b \) to have a common denominator: \[ P = \frac{2}{3}b + \frac{6}{3}b = \frac{8}{3}b \]

  5. Finding \( b \): Determine \( b \) when \( a \) is given as 10:

    • From \( a = \frac{1}{3}b \): \[ 10 = \frac{1}{3}b \] Multiply both sides by 3: \[ b = 30 \]

  6. Calculating the perimeter: Now use \( b = 30 \) in the perimeter formula: \[ P = \frac{8}{3}(30) = \frac{240}{3} = 80 \]

Conclusion:

So using the relationship \( a + b + a + b = 2a + 2b \), the perimeter of the pen is 80 yards.

i dont get it

Let’s break it down step by step in a clearer way, focusing on using the perimeter formula and the relationship between \( a \) and \( b \).

1. Understanding the Rectangle

  • The rectangle has sides:
    • Side \( a \) (length)
    • Side \( b \) (width)

2. Relationship Between Sides

You mentioned that \( a \) is connected to \( b \) by the equation: \[ a = \frac{1}{3}b \] This means that side \( a \) is one-third the length of side \( b \).

3. Perimeter Formula

The formula for the perimeter \( P \) of a rectangle is: \[ P = a + b + a + b \] This can be simplified to: \[ P = 2a + 2b \]

4. Substituting \( a \) in Terms of \( b \)

Now, replace \( a \) in the perimeter formula with the expression \( \frac{1}{3}b \): \[ P = 2\left(\frac{1}{3}b\right) + 2b \]

5. Simplifying the Perimeter Formula

Now let's simplify the expression:

  1. Calculate \( 2 \times \frac{1}{3}b \): \[ P = \frac{2}{3}b + 2b \]
  2. Convert \( 2b \) to a fraction with a denominator of 3:
    • \( 2b = \frac{6}{3}b \)
  3. Now combine the two terms: \[ P = \frac{2}{3}b + \frac{6}{3}b = \frac{8}{3}b \]

6. Finding \( b \) Given \( a = 10 \)

You mentioned using \( a = 10 \):

  • Using the relation \( a = \frac{1}{3}b \): \[ 10 = \frac{1}{3}b \] To find \( b \), multiply both sides by 3: \[ b = 10 \times 3 = 30 \]

7. Calculate the Perimeter

Now plug \( b = 30 \) back into the perimeter formula: \[ P = \frac{8}{3}(30) \] Calculating this gives: \[ P = \frac{240}{3} = 80 \]

Conclusion

Thus, the perimeter of A’Leila’s pen is 80 yards.

If you have any specific parts you want me to clarify further, please let me know!

WITHOUT ( { [

Got it! Let's write everything out clearly without any parentheses or brackets.

1. Relationship Between Sides

You have:

  • a = 1/3 b

2. Perimeter Formula

The formula for the perimeter P of a rectangle is: P = a + b + a + b This simplifies to: P = 2a + 2b

3. Substitute a into the Perimeter Formula

Now we will substitute a with (1/3)b: P = 2(1/3 b) + 2b

4. Simplify the Perimeter Formula

  • First, calculate 2(1/3 b): P = (2/3)b + 2b

  • Now convert 2b to a fraction: 2b = (6/3)b

  • Now combine the two terms: P = (2/3)b + (6/3)b P = (8/3)b

5. Finding b Given a = 10

Now, substitute a = 10 into the relationship: 10 = (1/3)b

  • Multiply both sides by 3: b = 10 * 3 b = 30

6. Calculate the Perimeter Using b = 30

Now plug b into the perimeter formula: P = (8/3)(30)

  • Now perform the multiplication: P = 240/3 P = 80

Conclusion

The perimeter of A’Leila’s pen is 80 yards.

If you need any further explanation or if there's anything else you'd like to clarify, feel free to ask!