Extended Distributive Property: Distributive property in geometry

Let's calculate the area of each rectangle step by step:

Step 1: Calculate the area of Rectangle A.
- Dimensions are $(x+4)$ and $(x+3)$.
- The area is calculated as $(x+4) \times (x+3)$.

Expanding this expression using the distributive property, we get:
$(x+4)(x+3) = x \cdot x + x \cdot 3 + 4 \cdot x + 4 \cdot 3$
$\Rightarrow x^2 + 3x + 4x + 12$
$\Rightarrow x^2 + 7x + 12$

Step 2: Calculate the area of Rectangle B.
- Dimensions are $(x+5)$ and $(x+2)$.
- The area is calculated as $(x+5) \times (x+2)$.

Using the distributive property to expand:
$(x+5)(x+2) = x \cdot x + x \cdot 2 + 5 \cdot x + 5 \cdot 2$
$\Rightarrow x^2 + 2x + 5x + 10$
$\Rightarrow x^2 + 7x + 10$

Step 3: Compare the areas of Rectangle A and B.
- Area of Rectangle A: $x^2 + 7x + 12$
- Area of Rectangle B: $x^2 + 7x + 10$

Subtract the area of Rectangle B from the area of Rectangle A:
$(x^2 + 7x + 12) - (x^2 + 7x + 10) = (x^2 + 7x + 12) - x^2 - 7x - 10$
$\Rightarrow x^2 - x^2 + 7x - 7x + 12 - 10$
$\Rightarrow 2$

Thus, the area of Rectangle A is larger by $2$ area units.

The correct answer, therefore, is: The area of rectangle A is larger by 2 area units.

To solve this problem, we apply the formula for the area of a trapezoid:

• Step 1: Identify and substitute the expressions for the bases and height:
  $b_1 = x + 5$, $b_2 = 13x - 2$, $h = 12x - 8$.
• Step 2: Calculate $b_1 + b_2$:
  $b_1 + b_2 = (x + 5) + (13x - 2) = 14x + 3$.
• Step 3: Substitute into the area formula:
  $A = \frac{1}{2} \times (14x + 3) \times (12x - 8)$.
• Step 4: Distribute and simplify:
  $A = \frac{1}{2} \times ((14x + 3) \cdot (12x - 8))$.
• Step 5: Expand the product:
  $(14x + 3)(12x - 8) = 14x \cdot 12x + 14x \cdot (-8) + 3 \cdot 12x + 3 \cdot (-8) = 168x^2 - 112x + 36x - 24$.
• Step 6: Combine like terms:
  $168x^2 - 76x - 24$.
• Step 7: Divide by 2 to find the area:
  $A = \frac{1}{2} \times (168x^2 - 76x - 24) = 84x^2 - 38x - 12$.

Therefore, the area of the trapezoid is $84x^2 - 38x - 12$.

To solve this problem, we'll follow these steps:

• Step 1: Calculate the area of the rectangle using the area formula for a rectangle.

• Step 2: Identify the conditions required for a valid rectangle by ensuring positive dimensions.

• Step 3: Analyze the provided choices to identify the correct answer.

Now, let's work through each step:

Step 1: The width of the rectangle is given as $-a + 3x$, and the height is $-5 + 4x$. The area of a rectangle is calculated by multiplying these two dimensions:

$\text{Area} = (-a + 3x)(-5 + 4x)$

Step 2: We'll expand the expression for the area:

$= (-a + 3x)(-5 + 4x) = (-a)(-5) + (-a)(4x) + (3x)(-5) + (3x)(4x)$

Step 3: Simplifying each term, we get:

$= 5a - 4ax - 15x + 12x^2$

Step 4: Reorganize the terms:

$= 12x^2 - 15x - 4ax + 5a$

Next, let's determine the conditions for the rectangle to exist, which means both dimensions must be positive:

• Width: $-a + 3x > 0 \implies 3x > a$

• Height: $-5 + 4x > 0 \implies 4x > 5 \implies x > \frac{5}{4} = 1\frac{1}{4}$

Therefore, the conditions for the rectangle to exist are $3x > a$ and $x > 1\frac{1}{4}$.

By evaluating the provided choices, we can see the correct choice is:

Area: $12x^2-15x-4ax+5a$

Conditions: $x > 1\frac{1}{4}$ and $3x > a$.

Thus, the correct choice is option 4. Confirming with the given correct answer, our solution matches perfectly.

To solve this problem, we'll calculate the area of the circle using the given radius expression. The process involves substituting and simplifying expressions:

• Step 1: Recognize that the area of a circle is given by the formula $A = \pi r^2$, where $r$ is the radius.
• Step 2: Substitute the given expression for the radius: $r = 7 - 5a$.
• Step 3: Calculate $r^2$ by expanding $(7 - 5a)^2$ using the identity for squaring a binomial.

Let's apply these steps:

First, substitute the expression for the radius into the area formula:
$A = \pi (7 - 5a)^2$.

Next, expand $(7 - 5a)^2$ using the distributive property or binomial expansion:
$(7 - 5a)^2 = 7^2 - 2 \cdot 7 \cdot 5a + (5a)^2 = 49 - 70a + 25a^2$.

Substituting back, we find:
$A = \pi (49 - 70a + 25a^2)$.

The area of the circle, simplified, is:
$A = 25\pi a^2 - 70\pi a + 49\pi$.

Therefore, the area of the circle in terms of $a$ is $25\pi a^2 - 70\pi a + 49\pi$.