Question

A thief plans to steal a gold sphere with a radius of 28.9 \(\mathrm{cm}\) from a museum. If the gold has a density of \(19.3 \mathrm{g} / \mathrm{cm}^{3},\) what is the mass of the sphere in pounds? [The volume of a sphere is \(V=(4 / 3) \pi r^{3} . ]\) Is the thief likely to be able to walk off with the gold sphere unassisted?

Short answer

The volume of the gold sphere is \(V = \frac{4}{3}\pi (28.9)^3 \approx 101216.087 cm^3\). The mass of the gold sphere is \(mass = 101216.087 \times 19.3 \approx 1953071.28 grams\), and in pounds, it is \(mass_{pounds} = \frac{1953071.28}{453.592} \approx 4304.55 pounds\). Since 4304.55 pounds is much higher than the comfortable lifting capacity of a healthy adult human (50 pounds), it is highly unlikely that the thief will be able to walk off with the gold sphere unassisted.

Step-by-step solution

Calculate the volume of the sphere

To find the volume of the sphere, we use the formula \(V = \frac{4}{3}\pi r^3\), where r is the radius of the sphere. The radius given is 28.9 cm, so we have: \[ V = \frac{4}{3}\pi (28.9)^3 \] Calculate the value of V.

Calculate the mass of the gold sphere

Now that we have the volume, we can calculate the mass of the gold sphere by multiplying the volume with its density, which is given as 19.3 g/cm³. \[mass = volume \times density\] \[mass = V \times 19.3 \] Insert the value of V from Step 1 and calculate the mass.

Convert the mass from grams to pounds

To convert the mass from grams to pounds, we use the conversion factor 1 pound = 453.592 grams. Divide the mass in grams by 453.592 to find its value in pounds: \[mass_{pounds} = \frac{mass_{grams}}{453.592} \] Insert the value of the mass in grams from Step 2 and calculate the mass in pounds.

Determine if the thief can carry the gold sphere unassisted

In this step, consider the mass of the gold sphere in pounds calculated in Step 3. On average, a healthy adult human can comfortably lift and carry a weight of about 50 pounds. If the mass of the gold sphere is significantly higher than this value, it is unlikely that the thief will be able to walk off with the gold sphere unassisted. Compare the calculated mass with this reference value and draw a conclusion about the thief's ability to carry the sphere.

Key concepts

Volume of a Sphere

Understanding how to calculate the volume of a sphere is crucial when determining the mass of a spherical object, such as a gold sphere. The volume can be described as the amount of space that the object occupies. The formula for the volume of a sphere is represented by \[ V = \frac{4}{3}\pi r^3 \]where \( V \) is the volume, \( \pi \) is a mathematical constant approximately equal to 3.14159, and \( r \) is the radius of the sphere—the distance from the center of the sphere to any point on its surface. This formula tells us that the volume of a sphere increases significantly with an increase in the radius, as the radius is cubed in the calculation. For example, if we were to double the radius, the volume would increase by a factor of eight, not two, because \( 2^3 = 8 \). This is why a seemingly small increase in the dimensions of a sphere can result in a large increase in volume, and subsequently, its mass when combined with material density.

Density

Density is a fundamental concept when it comes to determining the mass of an object. It is defined as the mass per unit volume of a substance. The denser a material, the more mass it contains in a given volume. In mathematical terms, this can be expressed as:\[density = \frac{mass}{volume}\]where the mass of the substance is measured in grams (g) and the volume in cubic centimeters (\(\mathrm{cm}^3\)). For instance, gold is known to have a high density, meaning a small volume of gold will have a significant mass. By knowing the density of gold, \(19.3 \mathrm{g}/\mathrm{cm}^3\), we can calculate the mass of a gold object once we know its volume. This step is essential when attempting to solve problems like estimating the mass of a gold sphere from its geometric measurements.

Unit Conversion

Converting units is a practical skill in many scientific calculations, including determining the mass of objects when the units provided in the problem differ from the desired units for the answer. For our example of a gold sphere's mass calculation, the final step involves converting mass from grams to pounds.

Such conversions require the use of conversion factors, which are standardized numeric ratios that allow mathematicians or scientists to convert measurements from one unit to another. The conversion factor to translate grams to pounds is:\[1 \text{pound} = 453.592 \text{grams}\]To convert the mass of an object from grams to pounds, we take the mass in grams and divide it by 453.592. The result is the mass expressed in pounds. This step is particularly important in our context, as the thief attempts to steal the gold sphere. Since most people are familiar with pounds as a measure of weight, converting the mass into pounds can help us visualize the heaviness of the object better and assess whether the thief can indeed walk away with the sphere unassisted.