Question

maria rents space at a farmers market to sell the vegetables she grows. she uses her own table that is 4 feet by 8 feet. raul lends her an extra table that is 3 feet wide. total table area = 3x + 8(4) total table area = 8(x + 4) - 5x why can both expressions be used to find the total table area? the expressions are both equivalent to 3x + 20, 3x + 32, 5x - 20, 5x - 32 (with a diagram of rental space and tables)

Explanation:

Step1: Simplify the first expression

The first expression for total table area is \(3x + 8(4)\). Calculate \(8(4)=32\)? Wait, no, wait. Wait, looking at the diagram, maybe I misread. Wait, the first table: the vertical table is 3 (width) and height \(x + 4 - 4=x\)? Wait, no, let's look at the second expression: \(8(x + 4)-5x\). Let's expand that: \(8x+32 - 5x=3x + 32\)? Wait, no, the first expression is \(3x+8(4)\). Wait, \(8(4)=32\), so \(3x + 32\)? But the dropdown has \(3x + 20\) selected. Wait, maybe I misinterpret the diagram. Let's re - examine:

The lower table is 8 feet long (since the total length is 8) and 4 feet wide, so its area is \(8\times4 = 32\). The upper table is 3 feet wide and \(x\) feet tall (since the height of the working space is \(x\), and the total height is \(x + 4\), so the upper table's height is \(x\)). Wait, no, the total height of the rental space is \(x + 4\), and the lower table is 4 feet tall, so the upper table is \(x\) feet tall. The upper table's area is \(3\times x=3x\). So total area is \(3x+32\). Now let's simplify the second expression: \(8(x + 4)-5x\). Expand \(8(x + 4)\): \(8x+32\). Then subtract \(5x\): \(8x + 32-5x=3x + 32\). Ah, so I made a mistake earlier. The first expression: \(3x+8(4)=3x + 32\) (since \(8\times4 = 32\)), and the second expression: \(8(x + 4)-5x=3x + 32\). So both expressions simplify to \(3x + 32\). Wait, but the dropdown has \(3x + 20\) selected. Wait, maybe the diagram is different. Wait, the working space has a width of 5 (since total width is 8, and the upper table is 3, so \(8 - 3=5\)) and height \(x\), and the lower table: wait, the lower table's width is 8 and height is 4, area \(8\times4 = 32\). The upper table: width 3, height \(x\), area \(3x\). But the second expression is \(8(x + 4)-5x\). Let's see: \(8(x + 4)\) is the area of the entire rental space (length 8, height \(x + 4\)), and then we subtract the working space, which is 5 (width) times \(x\) (height), so \(5x\). So total table area is rental space area minus working space area. Rental space area: \(8(x + 4)\), working space area: \(5x\), so table area: \(8(x + 4)-5x\). Now simplify \(8(x + 4)-5x\):

Step2: Simplify the second expression

Expand \(8(x + 4)\): \(8\times x+8\times4 = 8x + 32\). Then subtract \(5x\): \(8x+32 - 5x=(8x - 5x)+32=3x + 32\).

Now the first expression: \(3x+8(4)\). Calculate \(8(4)=32\), so \(3x + 32\).

So both expressions simplify to \(3x + 32\). Wait, but the dropdown in the problem has \(3x + 20\) selected. Maybe there's a misprint or misinterpretation. But according to the algebraic simplification:

First expression: \(3x+8(4)=3x + 32\)

Second expression: \(8(x + 4)-5x=8x + 32-5x=3x + 32\)

So both expressions are equivalent to \(3x + 32\) because when we simplify both, we get the same result. The first expression calculates the area of the two table parts (upper table: \(3x\), lower table: \(8\times4 = 32\)) and adds them. The second expression calculates the area of the entire rental space (\(8(x + 4)\)) and subtracts the area of the working space (\(5x\)) to get the table area. Since both simplifications lead to \(3x + 32\), they are equivalent and can both be used to find the total table area.

Answer:

The two expressions \(3x + 8(4)\) and \(8(x + 4)-5x\) can both be used to find the total table area because when simplified, they are equivalent.

• For the first expression \(3x+8(4)\):
• The term \(3x\) represents the area of the upper table (with width 3 and height \(x\)), and \(8(4)\) represents the area of the lower table (with length 8 and width 4). So the first expression is the sum of the areas of the two table sections.
• For the second expression \(8(x + 4)-5x\):
• The term \(8(x + 4)\) represents the area of the entire rental space (with length 8 and height \(x + 4\)). The term \(5x\) represents the area of the working space (with width 5 and height \(x\)). So the second expression is the area of the rental space minus the area of the working space, which also gives the area of the tables.

When we simplify both expressions:

• Simplifying \(3x+8(4)\): \(3x + 32\) (since \(8\times4 = 32\)).
• Simplifying \(8(x + 4)-5x\): First, expand \(8(x + 4)\) to get \(8x+32\), then subtract \(5x\): \(8x + 32-5x=3x + 32\).

Since both expressions simplify to \(3x + 32\), they are equivalent and can both be used to find the total table area. The correct equivalent expression from the dropdown (after correction) should be \(3x + 32\) (the initially selected \(3x + 20\) was likely an error).