Here are a few recommended readings before getting started with this lesson.
Dominika wants to buy a new aquarium for her fish. She is interested in two types of aquariums that have equal linear measurements.
Aquarium $A$ is a right cylinder with a diameter of $10$ feet and a height of $5$ feet. Additionally, a right cone sits inside of it. Aquarium $B$ is a hemisphere with a diameter of $10$ feet. Help Dominika answer the following questions.
Recall how the formula for the volume of a sphere is proven. The same thought process used in the proof can be applied to solve the challenge.
Two types of aquariums attract the attention of Dominica: Aquarium $A$ and Aquarium $B.$
Aquarium $A$ is a right cylinder with a diameter of $10$ feet and a height of $5$ feet. Additionally, its bottom base is a right cone. Aquarium $B,$ on the other hand, is a hemisphere with a diameter of $10$ feet.
Volume of the Cylinder | Volume of the Cone | |
---|---|---|
Formula | $V_{1}=πr_{2}h$ | $V_{2}=31 πr_{2}h$ |
Both the cone and the cylinder forming Aquarium $A$ have a $10$ foot diameter, and therefore both have a radius of $5$ feet. With that in mind, substitute $r=5$ and $h=5.$
Volume of the Cylinder | Volume of the Cone | |
---|---|---|
Formula | $V_{1}=πr_{2}h$ | $V_{2}=31 πr_{2}h$ |
Substitute Values | $V_{1}=π(5)_{2}5$ | $V_{2}=31 π(5)_{2}5$ |
Calculate | $V_{1}=125π$ | $V_{2}=3125π $ |
$V_{1}=125π$, $V_{2}=3125π $
$a=33⋅a $
Subtract fractions
Use a calculator
Round to nearest integer
Substitute values
Calculate power
Commutative Property of Multiplication
$(a−b)_{2}=a_{2}−2ab+b_{2}$
Distribute $-π$
Subtract term
$AB=5−h$, $AC=5$
$LHS−(5−h)_{2}=RHS−(5−h)_{2}$
Calculate power
$(a−b)_{2}=a_{2}−2ab+b_{2}$
Distribute $-1$
Subtract term
$LHS =RHS $
$a_{2} =a$
Cavalieri Principle |
Two solids with the same height and the same cross-sectional area at every altitude have the same volume. |
Both aquariums have a height of $5$ feet, and the area of the water’s surface when filled to a height of $h$ feet is the same for each aquarium.
Aquarium $A$ | Aquarium $B$ | |
---|---|---|
Height $(ft)$ | $5$ | |
Cross-Sectional Area $(ft_{2})$ | $10πh−πh_{2}$ |
Tiffaniqua wants to calculate the length of the a toilet paper roll. Hey! It is on a great sale, Okay. She draws a diagram and denotes the thickness of the paper, the inner radius, and the outer radius by $t,$ $r,$ and $R,$ respectively.
The area of the shaded region $A$ can be calculated in two ways. It can be expressed as the area of the circle of radius $R$ minus the area of the circle of radius $r.$ $A=πR_{2}−πr_{2} $
Alternatively, it can be expressed as the area of the front face of the long thin rectangular prism, which is created when the paper is unrolled. The area of the front face is the product of $t$ and $L.$ That product is also equal to $A.$ Therefore, $t⋅L$ can be substituted for $A$ in the derived formula for the area of the shaded region. Note that there might be several alternative ways to find the equation. Only one possible way was shown here.A cylindrical soda can is made of aluminum. It is $6$ inches high and its bases have a radius of approximately $1.2$ inches.
Give a go at answering the following set of questions. If necessary, round the answer to two decimal places.
$a⋅cb =ca⋅b $
Cross out common factors
Simplify quotient
Multiply
By modeling real-life objects using geometric shapes, various characteristics of the objects can be determined. These characteristics can then be compared to make inferences which could impact real decisions to be made.
Emily is attending a fair and wants to sell $1.5$ liters of homemade orange juice she is naming Oranjya Thirsty. She needs to decide the type of glass she will use to serve the juice — a cocktail glass or a Collins glass.
A cocktail glass is a type of glass that has an inverted cone bowl. The cone bowl's height is $5.8$ centimeters and the radius of its base is $5.4$ centimeters. A collins glass is a cylindrical glass with a height of $9$ centimeters and a radius of $3.2$ centimeters. Help Emily make a decision by answering the following questions.
$r=5.4$, $h=5.8$
Calculate power
Multiply
$b1 ⋅a=ba $
Use a calculator
Round to nearest integer
Type of Glass | |
---|---|
Cocktail Glass | Collins Glass |
$8⋅6$ | $5⋅10$ |
$$48$ | $$50$ |
About $1.7×10_{6}$
The formula for the volume of a sphere is $V=34 πr_{3},$ where $r$ is the radius of the sphere.
To find the number $N$ of grains of sand in Ramsha's hands, the mass $M$ of the sand in her hands should be divided by the mass $m$ of a grain. $N=mM $ Recall that the density of a substance is equal to its mass divided by its volume. In other words, the mass of a substance is its density times its volume. $d=Vm ⇔m=d⋅V $ Since the density of a grain of sand is given, the volume of a grain of sand will be calculated first.
$r=0.03$
Calculate power
Commutative Property of Multiplication
$ca ⋅b=ca⋅b $
Use a calculator
Round to ${\textstyle 5 \, \ifnumequal{5}{1}{\text{decimal}}{\text{decimals}}}$
Write in scientific notation
$ca ⋅b=ca⋅b $
Cancel out common factors
Simplify quotient
Commutative Property of Multiplication
Multiply
Round to ${\textstyle 1 \, \ifnumequal{1}{1}{\text{decimal}}{\text{decimals}}}$
$M=300g$, $m=1.7×10_{-4}g$
Cross out common factors
Simplify quotient
Write as a product of fractions
$a_{m}1 =a_{-m}$
Calculate quotient
Multiply
Round to $2$ significant ${\textstyle \ifnumequal{2}{1}{\text{digit}}{\text{digits}}}$
Write in scientific notation
the total number of stars in the universe is greater than all the grains of sand on all the beaches of the planet Earth.
Research projects usually require an interdisciplinary approach. That is, people from different disciplines work together to develop and test hypothesis, run experiments, and test theories.