Pregunta
upstudy study bank question image url

6. If M is replaced by one of the following numbers, the product \( (M-10)(M+14) \) becomes a perfect square. A. 0 B. 12 C. 14 D. 16 E. 18 7. In a triangle the three angles are in the ratio of \( 2: 3: 5 \). The difference between the largest and the smallest angle, in degrees = ? A. 0 B. 120 C. 90 D. 54 E. 72

Ask by Davies Gray. in South Africa
Mar 14,2025

Solución de inteligencia artificial de Upstudy

Respuesta verificada por el tutor

Responder

**Problem 6:** The value of \( M \) that makes \( (M-10)(M+14) \) a perfect square is **18**. **Problem 7:** The difference between the largest and smallest angle in the triangle is **54 degrees**.

Solución

**Problem 6** We are given the expression \[ (M-10)(M+14) \] and we need to find which value of \( M \) makes this product a perfect square. We test the choices: - **A. \( M=0 \)** \[ (0-10)(0+14) = (-10)(14) = -140 \] A negative number is not a perfect square (in the set of nonnegative integers). - **B. \( M=12 \)** \[ (12-10)(12+14) = (2)(26) = 52 \] \( 52 \) is not a perfect square. - **C. \( M=14 \)** \[ (14-10)(14+14) = (4)(28) = 112 \] \( 112 \) is not a perfect square. - **D. \( M=16 \)** \[ (16-10)(16+14) = (6)(30) = 180 \] \( 180 \) is not a perfect square. - **E. \( M=18 \)** \[ (18-10)(18+14) = (8)(32) = 256 \] \( 256 \) is equal to \( 16^2 \), which is a perfect square. Thus, the answer for Problem 6 is **E**. --- **Problem 7** The angles of a triangle are in the ratio \( 2:3:5 \). Let the measures of the angles be: \[ 2x, \quad 3x, \quad 5x \] Since the sum of the angles in a triangle is \( 180^\circ \), we have: \[ 2x + 3x + 5x = 180 \] \[ 10x = 180 \] \[ x = 18 \] Now, calculate each angle: \[ \text{Smallest angle} = 2x = 2(18) = 36^\circ \] \[ \text{Largest angle} = 5x = 5(18) = 90^\circ \] The difference between the largest and the smallest angle is: \[ 90^\circ - 36^\circ = 54^\circ \] Thus, the answer for Problem 7 is **54**, which corresponds to option **D**.

Respondido por UpStudy AI y revisado por un tutor profesional

error msg
Explicar
Simplifique esta solución

Extra Insights

In order for the expression \( (M-10)(M+14) \) to be a perfect square, we can substitute the given options for \( M \) and check the results. For instance, if we take \( M = 0 \), then the product becomes \( (-10)(14) = -140 \), which is not a perfect square. However, substituting \( M = 16 \) yields \( (16-10)(16+14) = 6 \times 30 = 180 \), once again not a perfect square. Playing around with values reveals \( M = 12 \) results in \( (12-10)(12+14) = 2 \times 26 = 52 \), leading to the conclusion that only \( M = 18 \) works since it perfectly squares \( 64 \). Triangles can be an interesting adventure in angles! Given the ratio of \( 2:3:5 \), we can let the angles be \( 2x, 3x, \) and \( 5x \). The total of angles in a triangle equals \( 180^\circ \). Setting up an equation: \( 2x + 3x + 5x = 180 \) gives us \( 10x = 180 \) or \( x = 18 \). Therefore, the angles are \(36^\circ, 54^\circ, \) and \(90^\circ\). The difference between the largest angle \(90^\circ\) and the smallest angle \(36^\circ\) is \(54^\circ\)!

preguntas relacionadas

¡Prueba Premium ahora!
¡Prueba Premium y hazle a Thoth AI preguntas de matemáticas ilimitadas ahora!
Quizas mas tarde Hazte Premium
Estudiar puede ser una verdadera lucha
¿Por qué no estudiarlo en UpStudy?
Seleccione su plan a continuación
Prima

Puedes disfrutar

Empieza ahora
  • Explicaciones paso a paso
  • Tutores expertos en vivo 24/7
  • Número ilimitado de preguntas
  • Sin interrupciones
  • Acceso completo a Respuesta y Solución
  • Acceso completo al chat de PDF, al chat de UpStudy y al chat de navegación
Básico

Totalmente gratis pero limitado

  • Solución limitada
Bienvenido a ¡Estudia ahora!
Inicie sesión para continuar con el recorrido de Thoth AI Chat
Continuar con correo electrónico
O continuar con
Al hacer clic en "Iniciar sesión", acepta nuestros términos y condiciones. Términos de Uso & Política de privacidad