Question

solution
3 the figure at the right has area 125 cm². which equation can be used to find the value of x?
a 125 = 30x - 5
b 35x = 125
c 25x = 125
d 125 ÷ 15 = 2x
josephine chose b as the correct answer. how might she have gotten that answer?

Explanation:

Step1: Find height of top rectangle

The total height of the figure is $15$ cm, and the height of the bottom rectangle is $2x$ cm. So the height of the top rectangle is $15 - 2x = 5$ cm. Solving for $2x$ gives $2x = 15 - 5 = 10$, so $2x = 10$.

Step2: Calculate total area

The figure is two rectangles. The bottom rectangle has area $2x \times x = 2x^2$. The top rectangle has area $5 \times x = 5x$. Total area: $2x^2 + 5x = 125$. Substitute $2x=10$ (so $x=5$) to verify, or express total area using the side lengths: the bottom rectangle is $2x$ by $x$, the top is $x$ by $5$. Total area can also be written as $x(2x + 5) = 125$. But since $2x=10$, substitute: $x(10 + 5)=15x=125$ is wrong, but to find the correct equation:
From $2x = 15 - 5$, $2x=10$, so $x=5$. Plug into total area: $2*5*5 +5*5=50+25=75$ no, wait correction: the bottom rectangle's height is $15 - 5 = 10 = 2x$, so bottom rectangle is $2x$ by $x$, top is $x$ by $5$. Total area: $2x*x + 5*x = 2x^2 +5x =125$. Factor: $x(2x+5)=125$. But $2x=10$, so $2x+5=15$, $15x=125$ is not correct, wait no: the total height is 15, so the height of the bottom rectangle is $15 - 5 = 10$, which equals $2x$, so $2x=10$, $x=5$. Then total area is $(2*5)*5 +5*5=50+25=75$ no, that's wrong. Wait, the bottom rectangle's side is $2x$, its height is $15 - 5 = 10$, so area is $2x*10$? No, no, the figure: bottom is a square? No, bottom is rectangle with length $2x$, height $15 - 5 = 10$, top is rectangle with length $x$, height $5$. Total area: $2x*10 + x*5 =20x +5x=25x=125$. That's option C.

Step3: Analyze Josephine's error

Josephine chose $35x=125$. She likely incorrectly added the lengths $2x + 5 + x = 3x$, then multiplied by $x$? No, more likely she added the heights $15 + 5 =20$? No, she probably miscalculated the total combined dimensions: she might have thought the total "length" for area is $2x + 5 + x = 3x$, and multiplied by $x$? No, or she incorrectly calculated the total area as $x(2x + 5 + 10)=x(2x+15)$ no, wait she got $35x=125$, so she probably added $2x + 5 + 10=35$? No, $2x + 5 + x + 10=3x+15$ no. Wait, she likely incorrectly assumed the total height of the figure is $x + 15$, and the width is $2x$, then subtracted nothing, but no. Alternatively, she added the two rectangle areas as $2x*15 +5*x=30x+5x=35x=125$, meaning she incorrectly calculated the bottom rectangle's height as 15 instead of $15-5=10$, so she used the full 15 cm as the height of the bottom rectangle instead of subtracting the 5 cm of the top rectangle.

Answer:

First, the correct equation is C. $25x = 125$

For Josephine's error: She likely incorrectly calculated the area of the bottom rectangle using the full total height of 15 cm (instead of $15 - 5 = 10$ cm) for its height. This would make her calculate total area as $(2x \times 15) + (x \times 5) = 30x + 5x = 35x$, leading her to the equation $35x = 125$.