Questions: El perímetro de un rectángulo es 128 cm. Si el largo equivale al triple del ancho. ¿Cuál es el área del rectángulo? a) 128 cm² b) 268 cm² c) 568 cm² d) 768 cm²

El perímetro de un rectángulo es 128 cm. Si el largo equivale al triple del ancho. ¿Cuál es el área del rectángulo? a) 128 cm² b) 268 cm² c) 568 cm² d) 768 cm²
Transcript text: El perímetro de un rectángulo es 128 cm . Si el largo equivale al triple del ancho. ¿Cuál es el área del rectángulo? a) $128 \mathrm{~cm}^{2}$ b) $268 \mathrm{~cm}^{2}$ c) $568 \mathrm{~cm}^{2}$ d) $768 \mathrm{~cm}^{2}$
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Solution

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Solution Steps

To find the area of the rectangle, we first need to determine the dimensions of the rectangle. We know the perimeter is 128 cm and the length is three times the width. Using the formula for the perimeter of a rectangle, we can set up an equation to solve for the width. Once we have the width, we can find the length and then calculate the area using the formula for the area of a rectangle.

Step 1: Definición de variables

Sea \( w \) el ancho del rectángulo. Dado que el largo es el triple del ancho, podemos expresar el largo como \( l = 3w \).

Step 2: Ecuación del perímetro

La fórmula del perímetro \( P \) de un rectángulo es: \[ P = 2(l + w) \] Sustituyendo el perímetro dado: \[ 128 = 2(3w + w) \] Simplificando la ecuación: \[ 128 = 2(4w) \implies 128 = 8w \]

Step 3: Resolución para el ancho

Despejando \( w \): \[ w = \frac{128}{8} = 16 \]

Step 4: Cálculo del largo

Sustituyendo el valor de \( w \) para encontrar \( l \): \[ l = 3w = 3 \times 16 = 48 \]

Step 5: Cálculo del área

El área \( A \) del rectángulo se calcula como: \[ A = l \times w = 48 \times 16 = 768 \]

Final Answer

La respuesta es \\(\boxed{768 \, \text{cm}^2}\\).

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