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Here are a few recommended readings before getting started with this lesson.
Tiffaniqua and Mark designed and 3D printed a drone. A neighbor hears about their feat and needs their help — she lost her dog Quipu! Before taking flight to find Quipo, first, they need to determine the flying conditions. The drone can travel at a maximum speed of 25 kilometers per hour. On a windy day, it travels 5 kilometers against the wind and then returns to the starting location.
Suppose that the drone constantly flies at its maximum rate throughout the search.
Tiffaniqua wants to put some of her math skils to use to help her parents. Her mom drives an SUV that travels 15 miles per gallon (mpg) and her father has a hybrid that travels 60 mpg. They travel the same distance every month. Tiffaniqua realizes that she can find a way to improve the combined miles per gallon. She is considering two options.
Option I | Tune the SUV to increase its mileage by 3 mpg and keep the hybrid as it is. |
---|---|
Option II | Buy a new hybrid that can travel 75 mpg and keep the SUV as it is. |
The combined mpg C is equal to the total miles divided by the total gallons. This can algebraically be written as C=SUV gallons+Hybrid gallonsSUV miles+Hybrid miles.
miles | mpg | gallons=mpgmiles | |
---|---|---|---|
SUV | x | 18 | 18x |
Hybrid | x | 60 | 60x |
Substitute expressions
Add terms
ba=b⋅10a⋅10
ba=b⋅3a⋅3
Add fractions
b/ca=ba⋅c
ba=b/xa/x
Calculate quotient
Round to 1 decimal place(s)
In this option, it is suggested to buy a new hybrid with a mpg of 75 and keep the SUV. As in the procedure followed in the previous option, the information at hand can be organized in a table.
miles | mpg | gallons=mpgmiles | |
---|---|---|---|
SUV | x | 15 | 15x |
Hybrid | x | 75 | 75x |
Substitute expressions
Add terms
ba=b⋅5a⋅5
Add fractions
b/ca=ba⋅c
ba=b/xa/x
Calculate quotient
If Option I is chosen, the combined mpg will be about 27.7. If Option II is chosen, it will be 25. Therefore, the first option gives a better combined mpg.
Tiffaniqua's parents want to thank her for solving their car problem and drove to a store 40 miles away in the next city over to buy a new 3D printer that she has dreamed of for years. On the way back, there was a road closure due to a landslide. This caused them to drive 12 mph slower.
This resulted in the return trip taking 3 hours longer. How many hours did it take them to get home from the store?Only the distance to the store is known. The time it takes to go to the store and the speed are unknown. Then, let t and r be the time elapsed and the speed, respectively.
Speed | Time | Distance | |
---|---|---|---|
To the store | r | t | 40 |
On the way home, Tiffaniqua's mother drove 12 mph slower and the trip took 3 hours longer.
Speed | Time | Distance | |
---|---|---|---|
To the store | r | t | 40 |
From the store | r−12 | t+3 | 40 |
(I): LHS/t=RHS/t
(II): LHS/(t+3)=RHS/(t+3)
a=tt⋅a
Subtract fractions
LHS⋅t=RHS⋅t
ca⋅b=ca⋅b
LHS⋅(t+3)=RHS⋅(t+3)
Distribute (t+3)
Distribute 40 & -12t
LHS−40t=RHS−40t
Commutative Property of Addition
Factor out -12
LHS/(-12)=RHS/(-12)
Use the Zero Product Property
(I): LHS+2=RHS+2
(II): LHS−5=RHS−5
Tiffaniqua and Mark are excited to print parts to make a drone using Tiffaniqua's new 3D printer. They want to make a special mixture for the raw material, called the filament. Tiffaniqua filled a flask with 500 milliliters of water. She then adds 60 grams of water soluble polyester resin.
Mark then adds more water at a rate of 10 milliliters per minute and simultaneously adds more resin at a rate of 6 grams per minute.
Tiffaniqua and Mark realize they can use their filament with Mark's old 3D printer in addition to the new printer. There is one major difference between the printers, however, it would take the old printer 50 minutes longer than the new printer to create the same parts.
If t is the printing time of the new printer, then t+50 is the printing time of the old printer.
LHS⋅200t(t+50)=RHS⋅200t(t+50)
Distribute 200t(t+50)
b1⋅a=ba
Cancel out common factors
Simplify quotient
Distribute 200 & t
Add terms
LHS−400t=RHS−400t
LHS−10000=RHS−10000
Rearrange equation
Use the Quadratic Formula: a=1,b=-350,c=-10000
-(-a)=a
Calculate power and product
Add terms
t=2350±162500 | |
---|---|
t=2350+162500 | t=2350−162500 |
t≈377 | t≈-27 |
t≈377
Write as a sum
Split into factors
1 h=60 min
Multiply
Drones communicate using radio waves on specific radio frequencies. The drone operates at a frequency of 2.4 gigahertz (GHz). How far a drone can travel depends on a number of factors such as the power of the signal transmitted by the controller.
Let Pt be the power of the radio signal transmitted by the controller, Pr the power of the radio signal received by the drone, λ the wavelength, and d the distance between the drone and its controller. It is known that Pr varies directly with Pt and the square of λ, and inversely with the square of d.
Substitute values
ca⋅b=a⋅cb
Calculate power
Multiply
Calculate quotient
Write in scientific notation
LHS/(7.5×10-5)=RHS/(7.5×10-5)
Write as a product of fractions
Calculate quotient
anam=am−n
a−(-b)=a+b
Calculate power
Multiply
Rearrange equation
Substitute values
Calculate power
Multiply
LHS⋅d2=RHS⋅d2
LHS/10-9=RHS/10-9
bma=ab-m
Multiply
LHS=RHS
a2=±a
Use a calculator
Round to nearest integer
From solving her parents' car issues, to successfully printing a drone, Tiffaniqua and Mark are now trying to locate a missing dog, Quipu! The owner tells them that Quipu was last seen running along the road that leads to the next city where Tiffaniqua bought the printer. They desperately want to find Quipo quickly. The drone can fly at a max speed of 25 kilometers per hour.
Before setting out to find Quipo, they test flew the drone 5 kilometers against the wind before needing to bring it back to the starting point. Suppose that the drone flew at its maximum rate of 25 km/h throughout the trip.
Upwind | Downwind | |
---|---|---|
Distance (km) | 5 | 5 |
Speed (km/h) | 25−w | 25+w |
Time (h) | 25−w5 | 25+w5 |
T(w)=