Pregunta
upstudy study bank question image url

A fence must be built to enclose a rectangular area of \( 20,000 \mathrm{ft}^{2} \). Fencing material costs \( \$ 4 \) per foot for the two sides facing north and south and \( \$ 8 \) per foot for the other two sides. Find the cost of the least expensive fence.

Ask by Willis Cook. in the United States
Mar 07,2025

Solución de inteligencia artificial de Upstudy

Respuesta verificada por el tutor

Responder

The least expensive fence costs \$3,200.

Solución

¡Inicia sesión para desbloquear respuestas gratis!

Una plataforma de aprendizaje en la que confían millones de estudiantes y profesores reales.

star-icon Descubrir

Respondido por UpStudy AI y revisado por un tutor profesional

Mind Expander

To find the least expensive fence enclosing a rectangular area of \( 20,000 \mathrm{ft}^{2} \), let \( x \) represent the length of the sides facing north and south, and \( y \) represent the length of the sides facing east and west. The area constraint gives us the equation: \[ x \cdot y = 20,000 \] From here, we can express \( y \) in terms of \( x \): \[ y = \frac{20,000}{x} \] Next, we calculate the cost of the fence. The total cost \( C \) can be expressed as: \[ C = 4(2y) + 8(2x) = 8y + 16x \] Substituting \( y \): \[ C = 8\left(\frac{20,000}{x}\right) + 16x \] \[ C = \frac{160,000}{x} + 16x \] To minimize this cost, we can take the derivative and set it to zero. The derivative \( \frac{dC}{dx} \) is: \[ \frac{dC}{dx} = -\frac{160,000}{x^2} + 16 \] Setting the derivative to zero: \[ -\frac{160,000}{x^2} + 16 = 0 \] \[ \frac{160,000}{x^2} = 16 \] \[ x^2 = \frac{160,000}{16} = 10,000 \] \[ x = 100 \] Now we find \( y \): \[ y = \frac{20,000}{100} = 200 \] So, the dimensions of the rectangle that minimizes the cost are \( x = 100 \mathrm{ft} \) and \( y = 200 \mathrm{ft} \). Now, we can calculate the cost of the fence: \[ C = 8(200) + 16(100) = 1600 + 1600 = 3200 \] Thus, the cost of the least expensive fence is \( \boxed{3200} \). Let’s put on our history hats! Did you know that fencing dates back to ancient civilizations? The Egyptians, for instance, utilized simple enclosures for their gardens and livestock. Fast forward to today, the fencing industry has evolved into a billion-dollar sector with a vast range of materials and designs! If you’re looking to tackle fencing projects, it’s crucial to check local building codes and regulations. Sometimes, what seems like an easy DIY project can land you in hot water if you unintentionally trespass into zoning issues or neighbor disputes! Always do thorough research and, when in doubt, consult with a local expert.

¡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