Question

an architect plans to build an extension to meilings rectangular deck. let x represent the increase, in meters, of her decks length. the expression 3(x + 6) represents the area of the deck, where 3 is the width, in meters, and (x + 6) represents the extended length, in meters. use the distributive property to write an expression that represents the total area of meilings new deck.
click the icon to view the deck.
an expression that represents the total area of meilings new deck is
(simplify your answer. use integers or decimals for any numbers in the expression)

Explanation:

Step1: Identify the formula for area of a rectangle

The area \( A \) of a rectangle is given by the formula \( A = \text{length} \times \text{width} \). Here, the width is \( 3(x + 6) \) and the length is \( (x + 6) \)? Wait, no, wait. Wait, the problem says: "The expression \( 3(x + 6) \) represents the area of the deck, where 3 is the width, in meters, and \( (x + 6) \) represents the extended length, in meters." Wait, no, wait, the problem says: "An architect plans to build an extension to Meiling’s rectangular deck. Let \( x \) represent the increase, in meters, of her deck’s length. The expression \( 3(x + 6) \) represents the area of the deck, where 3 is the width, in meters, and \( (x + 6) \) represents the extended length, in meters. Use the Distributive Property to write an expression that represents the total area of Meiling’s new deck." Wait, maybe I misread. Wait, no, the area of a rectangle is length times width. So if width is 3 and length is \( (x + 6) \), but wait, maybe the original deck? No, the problem is to find the total area, which is width times length. Wait, the width is 3, and the length is \( (x + 6) \)? Wait, no, the expression \( 3(x + 6) \) is the area? Wait, no, the problem says "the expression \( 3(x + 6) \) represents the area of the deck, where 3 is the width, in meters, and \( (x + 6) \) represents the extended length, in meters." Wait, that can't be, because area is length times width, so if width is 3 and length is \( (x + 6) \), then area is \( 3 \times (x + 6) \), which is \( 3(x + 6) \). But the problem says "Use the Distributive Property to write an expression that represents the total area". Wait, maybe the width is 3 and the length is \( (x + 6) \), so to expand \( 3(x + 6) \) using the distributive property (which is \( a(b + c) = ab + ac \)). So applying the distributive property to \( 3(x + 6) \):

Step2: Apply the Distributive Property

The Distributive Property states that \( a(b + c) = ab + ac \). Here, \( a = 3 \), \( b = x \), and \( c = 6 \). So:

\( 3(x + 6) = 3 \times x + 3 \times 6 \)

Step3: Simplify the products

Calculate \( 3 \times x = 3x \) and \( 3 \times 6 = 18 \). So combining these, we get:

\( 3x + 18 \)

Answer:

\( 3x + 18 \)