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Chapter 11: Problem 89

A water bed for sale has dimensions of 1.83 \(\mathrm{m} \times 2.13 \mathrm{m} \times 0.229 \mathrm{m} .\) The floor of the bedrroom will tolerate an additional weight of no more than 66660 \(\mathrm{N}\) . Find the weight of the water in the bed and determine whether the bed should be purchased.

Short Answer

Expert verified
The water in the bed weighs approximately 8738 N, and the bed can be safely purchased.

Step by step solution

01

Calculate the Volume of the Water Bed

First, find the volume of the water bed using the formula for the volume of a rectangular prism: \[V = ext{length} \times ext{width} \times ext{height}\]where:- length = 1.83\,\mathrm{m}- width = 2.13\,\mathrm{m}- height = 0.229\,\mathrm{m}By substituting these values:\[V = 1.83 \times 2.13 \times 0.229 = 0.891654\,\mathrm{m}^3\]
02

Calculate the Weight of the Water

Next, calculate the weight of the water using the density of water, which is approximately \(1000\,\mathrm{kg/m}^3\), and the gravitational force \(g = 9.8\,\mathrm{m/s}^2\). The weight \(W\) is given by the formula:\[W = V \times ext{density} \times g\]By substituting the known values:\[W = 0.891654 \times 1000 \times 9.8 = 8738.2092\,\mathrm{N}\]
03

Compare the Weight to the Maximum Tolerable Weight

Finally, compare the calculated weight of the water to the maximum tolerable weight of the floor, which is \(66660\,\mathrm{N}\). Since the weight of the water \(8738.2092\,\mathrm{N}\) is much less than \(66660\,\mathrm{N}\), the floor can support the water bed comfortably.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Volume Calculation
To calculate the volume of an object, especially one shaped like a rectangular prism, you need three main measurements:
  • Length
  • Width
  • Height
These dimensions are multiplied together using the formula: \[ V = \text{length} \times \text{width} \times \text{height} \]This formula gives you the volume in cubic meters if the dimensions are in meters. For the water bed in question, we have:
  • Length: 1.83 meters
  • Width: 2.13 meters
  • Height: 0.229 meters
By substituting these into the formula, we calculate:\[ V = 1.83 \times 2.13 \times 0.229 = 0.891654 \, \text{m}^3 \]This means the water bed can hold approximately 0.891654 cubic meters of water, which is the total amount of space inside the bed.
Density of Water
Density is a key concept in understanding how much something weighs based on its volume. The density of an object is defined as its mass per unit volume. For water, this density is:\[ \text{Density of water} = 1000\,\text{kg/m}^3 \]This means that 1 cubic meter (\(\text{m}^3\)) of water has a mass of 1000 kilograms. Knowing the volume of the water (in this case, the volume of the water bed), you can determine the mass of the water by multiplying the volume and the density:\[ \text{Mass of water} = V \times \text{Density} \]For our water bed, with a volume of 0.891654 \(\text{m}^3\), the mass would be:\[ \text{Mass} = 0.891654 \times 1000 = 891.654\,\text{kg} \]This calculation helps us know how much mass, and eventually how much weight, the water will have when considering gravitational effects.
Gravitational Force
Gravitational force or gravity is the force that pulls objects towards each other, such as how the Earth pulls objects towards its center. The gravitational force on an object at the Earth's surface can be calculated using:\[ g = 9.8\,\text{m/s}^2 \]To find the weight of an object, you multiply its mass by the gravitational acceleration (\(g\)). The formula for weight is:\[ W = m \times g \]For the water bed, once we have the mass of the water (891.654 kg), the weight can be calculated by:\[ W = 891.654 \times 9.8 = 8738.2092\,\text{N} \]This result tells us the force exerted by the water due to gravity. In this case, the floor must support this force without collapsing. Since 8738.21 N is less than the floor's tolerable weight of 66660 N, the bed is safe to use.

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