Question

the figure shown has a total area of 420 m². which equation can be used to find the value of x? 420 = 25·x + 15·3x 420 = 25·x + 15·2x 420 = 25·15 + 3x·x 420 = 25·4x + 15·3x

Explanation:

Step1: Analyze the figure's area composition

The L - shaped figure can be divided into two rectangles. One rectangle has dimensions \(25\) m (height) and \(x\) m (width). The other rectangle: the height of this rectangle is \(15\) m, and the width: since the total height of the left - most part is \(25\) m and the lower part is \(15\) m, the vertical length of the upper part of the L - shape is \(25 - 15=10\) m? Wait, no, looking at the horizontal side, the right - hand side horizontal length is \(3x\) m, and the left - hand side vertical part has width \(x\) m. Wait, actually, the second rectangle (the lower right part) has a height of \(15\) m and a width of \(3x\) m? No, wait, let's re - examine.

Wait, the vertical side of the left rectangle is \(25\) m and width \(x\) m. The other rectangle: the height is \(15\) m, and the width: the total horizontal length of the lower part. Wait, the upper horizontal segment is \(3x\) m, and the left vertical segment has width \(x\) m. Wait, the height of the upper rectangle (the part that sticks up) is \(25 - 15 = 10\) m? No, maybe a better way: the area of the L - shape is the sum of two rectangles.

First rectangle: height \(25\) m, width \(x\) m, area \(25\times x\).

Second rectangle: height \(15\) m, width \(3x\) m? No, wait, the vertical difference between \(25\) and \(15\) is \(10\) m, but the horizontal length of the lower part: the right - hand horizontal length is \(3x\) m, and the left - hand vertical part has width \(x\) m. Wait, no, let's look at the options.

Wait, the correct way: the figure can be split into two rectangles. One rectangle is \(25\) m (height) by \(x\) m (width). The other rectangle: the height is \(15\) m, and the width: since the upper part of the L - shape (the vertical part) has width \(x\) m, and the right - hand horizontal part has length \(3x\) m? No, wait, the second rectangle's width: the total horizontal length of the lower part. Wait, the vertical side of the left rectangle is \(25\) m, and the lower rectangle (the one with height \(15\) m) has a width of \(3x\) m? No, let's check the options.

Wait, the first option is \(420 = 25\cdot x+15\cdot3x\), the second is \(420 = 25\cdot x + 15\cdot2x\), etc.

Wait, let's calculate the area of each part. The left rectangle: height \(25\) m, width \(x\) m, area \(A_1 = 25x\). The right - hand rectangle: height \(15\) m, and the width: looking at the horizontal side, the length is \(3x\) m? Wait, no, the vertical length of the left rectangle is \(25\) m, and the vertical length of the right rectangle is \(15\) m. The horizontal length of the right rectangle: the upper horizontal segment is \(3x\) m, and the left vertical segment has width \(x\) m. Wait, maybe the width of the right rectangle is \(3x\) m. Wait, no, let's think about the dimensions.

Wait, the height of the left rectangle is \(25\) m, width \(x\) m. The height of the right rectangle is \(15\) m, and the width is \(3x\) m? No, that can't be. Wait, the vertical difference between \(25\) and \(15\) is \(10\) m, but the horizontal length of the lower part: the right - hand horizontal length is \(3x\) m, and the left - hand vertical part has width \(x\) m. Wait, maybe the width of the second rectangle is \(3x\) m. Wait, let's check the area formula.

The total area is the sum of the areas of the two rectangles. The first rectangle: area \(=25\times x\). The second rectangle: the height is \(15\) m, and the width: since the upper part of the L - shape (the part that is above the \(15\) m height) has a vertical length of \(25 - 15=10\) m, but the horizontal length of the lower part (the \(15\) m height part) is \(3x\) m? No, wait, the correct split: the L - shape can be divided into a rectangle with length \(25\) and width \(x\), and another rectangle with length \(15\) and width \(3x\). So the total area \(A=25x + 15\times3x\), and the total area is \(420\) \(m^2\). So the equation is \(420 = 25\cdot x+15\cdot3x\).

Step2: Verify the other options

- For the second option \(420 = 25\cdot x+15\cdot2x\): The width of the second rectangle is not \(2x\), so this is incorrect.
- For the third option \(420 = 25\cdot15 + 3x\cdot x\): The area of the first part is not \(25\times15\) (because the left rectangle has width \(x\), not \(15\)), and the second part is not \(3x\times x\), so this is incorrect.
- For the fourth option \(420 = 25\cdot4x+15\cdot3x\): The width of the first rectangle is not \(4x\), so this is incorrect.

Answer:

\(420 = 25\cdot x+15\cdot3x\) (the first option among the given options)