Question

the construction crew for a new store will dig a hole in the shape of a rectangular prism for the stores foundation. the hole will be 14 yd long, 12 yd wide, and 3 yd deep. to haul away all of the dirt from the hole, a dump truck will take trips to a nearby landfill. the crew knows the amount of dirt the truck can hold each trip in cubic feet.
(a) find the volume of dirt that will be hauled away in cubic feet. use the table of conversion facts, as needed.
3
ft
(b) each trip, the truck can haul away 335 ft³ of dirt. to save gas, the crew wants to take as few trips as necessary to haul all of the dirt away. since all of the dirt needs to be hauled away, the crew might have to take a partial load for one trip. how many trips are needed to haul away all of the dirt? count each trip it makes the same, even if the truck isnt full.
trips
(c) if it costs $1275 for each trip that the truck makes, how much will it cost to haul away all the dirt?
$
conversion facts for length
1 foot (ft) = 12 inches (in)
1 yard (yd) = 3 feet (ft)
1 yard (yd) = 36 inches (in)

Explanation:

(a)

Step1: Convert yard to feet

La longitud, ancho y profundidad están en yardas. Sabemos que \( 1 \, \text{yd} = 3 \, \text{ft} \). Entonces:

• Longitud: \( 14 \, \text{yd} \times 3 = 42 \, \text{ft} \)
• Ancho: \( 12 \, \text{yd} \times 3 = 36 \, \text{ft} \)
• Profundidad: \( 3 \, \text{yd} \times 3 = 9 \, \text{ft} \)

Step2: Calcular el volumen del prisma rectangular

El volumen \( V \) de un prisma rectangular se calcula con la fórmula \( V = l \times w \times h \), donde \( l \) es la longitud, \( w \) el ancho y \( h \) la altura (profundidad). Sustituyendo los valores:
\( V = 42 \, \text{ft} \times 36 \, \text{ft} \times 9 \, \text{ft} \)
\( V = 42 \times 36 \times 9 \)
Primero, \( 42 \times 36 = 1512 \)
Luego, \( 1512 \times 9 = 13608 \, \text{ft}^3 \)

Step1: Dividir el volumen total entre el volumen por viaje

El volumen total de la tierra es \( 13608 \, \text{ft}^3 \) y el camión puede transportar \( 335 \, \text{ft}^3 \) por viaje. Entonces, el número de viajes \( n \) se calcula como \( n = \frac{13608}{335} \)
Calculando: \( 13608 \div 335 \approx 40.62 \)
Como no se puede hacer un viaje parcial (se cuenta cada viaje, incluso si no está lleno), se redondea hacia arriba a \( 41 \) viajes.

Step1: Calcular el costo total

Cada viaje cuesta \( \$1275 \) y se necesitan \( 41 \) viajes. Entonces, el costo total \( C \) es \( C = 1275 \times 41 \)
Calculando: \( 1275 \times 41 = 1275 \times (40 + 1) = 1275 \times 40 + 1275 \times 1 = 51000 + 1275 = 52275 \)

Answer:

\( 13608 \, \text{ft}^3 \)

(b)