Look at the rectangle in the figure below. What is its area?
What do a and x need to be for the rectangle to exist?
Look at the rectangle in the figure below. What is its area?
What do a and x need to be for the rectangle to exist?
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 , and the height is . The area of a rectangle is calculated by multiplying these two dimensions:
Step 2: We'll expand the expression for the area:
Step 3: Simplifying each term, we get:
Step 4: Reorganize the terms:
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:
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.
Area:
Conditions:
x > 1\frac{1}{4}
3x>a