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Last weekend Ali visited an aviation festival near his neighborhood. In an experiment at the festival, an engineer introduced the element rhenium, which is used to make jet engine parts. The engineer showed the onlookers a small pyramid-shaped part of the element. The part was obtained by cutting one corner of a cube with edges of length 6 centimeters as shown.
What is the density of rhenium?Ali wonders.
The concept of density can be considered in different contexts. Density is essentially a derived unit that compares a quantity per unit of area or volume.
Maya is running a population census for her home city. She estimates the number of people in a region with a 4-mile radius to be around 220000.
Find the population density using the unit measurement of people per square mile. Then, round the answer to the nearest integer. Be sure to explain what this number means.Population Density: About 4376 people per square mile
Explanation: See solution.
The population density is the ratio of the number of people living in a region to region's area.
Substitute values
In physics, density refers to the ratio of the mass of a substance to its volume. The next few examples are about the density based on volume.
Izabella is planning a to make a nice lunch, she buys a bottle of water and a bottle of cooking oil. She knows that both bottles have a volume of 5 liters. st1d=cm30.92g, V=5000cm3
ca⋅b=ca⋅b
Cancel out common factors
Simplify quotient
Multiply
Bees build their honeycombs in such a way that each cell is a prism with a regular hexagonal base. In these small hexagonal cells, bees are born and raised, and honey and pollen are stored.
Diego, a bee-loving biology student, discovers that the depth of a hexagonal cell is 0.5 centimeters, and its base has a side length of 0.3 centimeters. Help Diego answer his following research questions.
Substitute values
LHS⋅s=RHS⋅s
a⋅cb=ca⋅b
Rearrange equation
B=0.23, h=0.5
Multiply
Round to 2 decimal place(s)
Substitute values
LHS⋅5=RHS⋅5
Rearrange equation
Use a calculator
Round to nearest integer
As the diagram indicates, the base of the hollow part is a square with a side length of 5 centimeters. Additionally, the mold is a cube with an edge length of 7 centimeters. Therefore, the hollow is a square prism with a base edge length of 5 centimeters and height of 7 centimeters.
The volume of this square prism can be calculated as the product of its base area and its height. Since the base of the prism is a square with a side length of 5 centimeters, the base area is 52 square centimeters. The volume of the hollow is 175cm3.The diagram shows a water tank that is positioned horizontally with some water inside. The water level height is 10 inches, and the distance between two bases is 50 inches.
Kevin wonders how high the water level will rise when the tank is positioned vertically. Help Kevin to find it by answering the following questions.
Ab=π(20)2
Calculate power
b1⋅a=ba
Since the height of a solid is the distance between its bases, multiplying Ab by 50 will give the volume of the portion of the cylinder.
As=3400π, h=50
ca⋅b=ca⋅b
Use a calculator
Round to nearest integer
At=1003, h=50
Use a calculator
Round to nearest integer
Vw=12284, Ab=π(20)2
Calculate power
Commutative Property of Multiplication
LHS/400π=RHS/400π
Rearrange equation
Use a calculator
Round to nearest integer
After discussing hexagonal cells of beehives, Diego thinks he can approximate the number of cells in his body. His teacher recommends for Diego to treat a cell like a sphere with a diameter of 2×10-3 centimeters.
If Diego's weight is 60 kilograms and the density of a cell is approximately the density of water, which is 1 gram per cubic centimeter, help Diego approximate the number of cells in his body. Write the answer in scientific notation.
About 1.4×1013 cells
The formula for the volume of a sphere is V=34πr3, where r is the radius of the sphere.