module 4
The purity of gold is measured in carats. 24-carat gold is pure gold, and has a density of 19.3 g/cm³. 18-carat gold is often used for jewelry because it holds its shape better than pure gold. An 18-carat gold ring, which is 75% pure gold, has a mass of 18 g. What volume of pure gold was used to make the ring, to the nearest hundredth?
18 g x 75%= the mass of 24-carat gold do mass of 24-carat gold/the density of 24-carat gold to find the volume
three dimensional figures affected by scale factor a
SA: a² V: a³
Population Density
number of people living per mi² (or per km²) population/area
two dimensional figures affected by scale factor a
perimeter: a area: a²
volume of a cone
1/3πr²h
Modeling to Meet Constraints
-a full-grown tree needs to have a minimum size canopy to photosynthesize enough sugar to feed the tree's bulk -this constraint can be modeled by relating the tree's canopy surface area to the volume of its trunk.
British Thermal Unit (BTU)
-a unit of energy, is approximately the amount of energy needed to increase the temperature of one pound of water by one degree Fahrenheit -energy content of a fuel may be measured in BTUs per unit of volume
Density
-the amount of matter that an object has in a given unit of volume -the density of an object is calculated by dividing its mass by its volume D=m/V
The density of water at 4 C is 1000 kg/m³. A cubic meter of water, when frozen to -20 C, has a volume of 1.0065 m³. What is the density of ice at this temperature, to the nearest tenth?
-the mass doesn't change -so do 1000/1.0065
Determining Dimensions given a volume
-volume formulas are useful for solving problems where the constraint is to use a given volume of material for a given shape -for instance, suppose you want to make a cylindrical candle using a given amount of wax -you can use the formula for the volume of a cylinder to determine the candle's dimensions
Assume a full-grown oak tree requires at least 8 ft² of exterior canopy area per cubic foot of trunk volume. Model the canopy with a hemisphere, and model the trunk using a cylinder whose height is three times its diameter. What is the minimum radius of canopy required for an oak with trunk diameter of 9 ft?
1) Find the volume of of the trunk by doing π(4.5)²(27) 2) Then multiply the volume by the minimum required (ft² of exterior canopy area/cubic foot of tree volume) 3) once you find the area of the exterior canopy use the surface area formula of a sphere, but halved since it is a hemisphere 4) then solve for the radius
A cylindrical space station is 5 m in diameter and 12 m long, and it requires 0.2 m² of solar panels per cubic meter of volume to provide power. If it has two sets of rectangular solar panels, each 2 m wide, how long should each set of panels be?
1) Find the volume of the cylinder π(2.5)²(12) 2) then multiply the volume by 0.2 so you get the area of both solar panels 3) then divide it by two so you find the area of each of the solar panels 4) since you know they are 2 m wide do 2 (x), x being the length and make that equal to the area of one solar panel and solve for x to get the length
A roll of aluminum foil is 15 in wide. It has an interior diameter of 1.2 in. and an exterior diameter of 1.6 in. If the foil is 0.001 in. thick, what length of foil is rolled up, to the nearest foot? (Hint: Start by finding the volume of a 1-ft length of foil 15 in. wide)
1) Find the volume of the foil by doing volume exterior cylinder ((0.8)²(15)π) - interior cylinder ((0.6)²(15)π) 2) then make a little cut of 1 foot of foil. Since you know the length is 12 in (because 1 foot) and the width is 15 in and the thickness is 0.001. That way you can find the volume per 1 foot length 3) once you find its volume/foot do (the volume of foil)(1 foot length/volume) to get the length.
volume of a pyramid
1/3Bh