PROBLEM #1, SET 2 - NUMBERS

1. How deep would the water in a trench three inches wide and sixteen feet long have to be to contain 24.0 cubic feet of water?

This problem is likely best done with W5P method because it asks for a length when you are given two other lengths and a volume. All units are in English system.

GIVEN:   l1 = 16 ft.,       l2 = 3 in,       l = 24.0 ft3
You know this from the units and dimensions table in the Units section of Lessons.

FIND:   a length, l
The question asks, "how deep," which asks for a length. You know the symbol for length from the units and dimensions table in the Units section of Lessons.

FORMULA:   V = l l l
You know two lengths and a volume. You are asked to find a length. The volume formula must be the one you need because it is the relationship among three lengths and a volume. You know this from the list of formulas in the formula section of Units in Lessons.

SOLVE: Using algebra, solve for the length you are looking for. Once you have one of the lengths on one side alone, substitute the known values from GIVEN into the working equation, simplify, and do the math. Here the math includes some DA to get the units right.

WHAT YOU WRITE DOWN SHOULD LOOK LIKE THIS:

GIVEN:   l1 = 16 ft.,       l2 = 3 in,       V = 24.0 ft3
FIND:   a length, l

The calculator math is:   24 ÷ 16 ÷ 3 x 12 =

ANSWER: Calculate the numerical answer, show your cancellation of the units, and show the raw answer (the units that remain and the number on your calculator). Change the raw answer to the correct number of significant digits and change into scientific notation if the answer is greater than 1000 or less than 1/1000. In this case, no change is needed.

PROBLEM #2, SET 2 - NUMBERS

2. Cloth is four feet wide as it comes on the bolt. What length of cloth would you need to cover a surface of 58.0 square yards?

This problem is likely best done with W5P method because it asks for a length when you are given one other length and an area. All units are in English system.

GIVEN:   l1 = 4 ft.,     A = 58.0 ft2
You know this from the units and dimensions table in the Units section of Lessons.

FIND:   a length, l
The question asks, "what length," which asks for a length. You know the symbol for length from the units and dimensions table in the Units section of Lessons.

FORMULA:   A = l l
You know one length and an area. You are asked to find a length. The area formula must be the one you need because it is the relationship among two lengths and a volume. You know this from the list of formulas in the formula section of Units in Lessons. State the original formula here.

SOLVE: Using algebra, solve for the length you are looking for. Once you have one of the lengths on one side alone, substitute the known values from GIVEN into the working equation, simplify, and do the math. Here the math includes some DA to get the units right.

WHAT YOU WRITE DOWN SHOULD LOOK LIKE THIS:
GIVEN:   l1 = 4 ft.,     A = 58.0 ft2
FIND:   a length, l

The calculator math is:   58 ÷ 4 x 3 =

ANSWER: Calculate the numerical answer, show your cancellation of the units, and show the raw answer (the units that remain and the number on your calculator). Change the raw answer to the correct number of significant digits and change into scientific notation if the answer is greater than 1000 or less than 1/1000. No change to the raw answer is needed in this problem.

PROBLEM #3, SET 2 - NUMBERS

3. It is 762 miles from here to Chicago. An obese Chemistry teacher jogs at a rate of one mile every twenty minutes. How long would it take him to jog to Chicago if he jogged continuously and REALLY wanted to get there?

This problem is likely best done with W5P method because it asks for a time when you are given a length and a velocity. All units are in English system.

GIVEN:   d* = 762 miles,     v = 1 mile/20 min
You know this from the units and dimensions table in the Units section of Lessons.
* I will admit to some sneakiness here. The dimension of distance has more than one symbol. Seeing the velocity and distance in the problem, I chose the symbol for distance that goes with the formula.

FIND:   a time, t
The question asks, "How long would it take ...," which asks for a time. You know the symbol for time from the units and dimensions table in the Units section of Lessons.

FORMULA:   v t = d
You know a length and a velocity. You are asked to find a time. The rate equation must be the one you need because it is the relationship among a rate (or velocity), a time, and a distance. You know this from the list of formulas in the formula section of Units in Lessons.

SOLVE: Using algebra, solve for the length you are looking for. Once you have the time on one side alone, substitute the known values from GIVEN into the working equation, simplify, and do the math. Here the math includes some DA to get the time units manageable.

WHAT YOU WRITE DOWN SHOULD LOOK LIKE THIS:

GIVEN:   d = 762 miles,     v = 1 mile/20 min

FIND:   a time, t

The calculator math is:   762 x 20 ÷ 60 =

ANSWER: Calculate the numerical answer, show your cancellation of the units, and show the raw answer (the units that remain and the number on your calculator). Change the raw answer to the correct number of significant digits and change into scientific notation if the answer is greater than 1000 or less than 1/1000. Here, because of the W5P, there is no need to change the raw answer.

PROBLEM #4, SET 2 - NUMBERS

4. The density of lead is 11.4 g/cc. What is the mass of a block of lead two cm by ten cm by four cm?

This problem is likely best done with W5P method because it asks for a mass when you are given three lengths and a density. All units are in metric system.

GIVEN:   l1 = 2 cm,     l2 = 10 cm,     l3 = 4 cm,     D = 11.4 g/cc
You know this from the units and dimensions table in the Units section of Lessons.

FIND:   a mass, m
The question asks, "What is the mass ...," which asks for m, the mass. You know the symbol for mass from the units and dimensions table in the Units section of Lessons.

FORMULAS:   V = l l l     D = m/V
You know three lengths and a density. You are asked to find a mass. For this question, you need two of the equations from the formula section of Units in Lessons, the formula for volume and the density formula.

SOLVE: Using algebra, solve for the length you are looking for. Once you have the time on one side alone, substitute the known values from GIVEN into the working equation, simplify, and do the math. Here the math includes some DA to get the units right.

WHAT YOU WRITE DOWN SHOULD LOOK LIKE THIS:

GIVEN:   l1 = 2 cm,     l2 = 10 cm,     l3 = 4 cm,     D = 11.4 g/cc

FIND:   a mass, m

The calculator math is:   11.4 x 2 x 10 x 4 =

ANSWER: Calculate the numerical answer, show your cancellation of the units, and show the raw answer (the units that remain and the number on your calculator). Change the raw answer to the correct number of significant digits and change into scientific notation if the answer is greater than 1000 or less than 1/1000. Once again there is no need to change the raw answer.

PROBLEM #5, SET 2 - NUMBERS

5. How many liters of water could you fit into a container measuring 10.0 ft by 13.0 ft by 8.00 ft?

This problem is likely best done with W5P method because it asks for a volume when you are given three lengths. The answer is asked for in metric system, but the information has been given in English system, so the units will need some DA.

GIVEN:   l1 = 10.0 ft,     l2 = 13.0 ft,     l3 = 8.00 ft,
You know this from the units and dimensions table in the Units section of Lessons.

FIND:   a volume, V
The question asks for a volume, V. You know the symbol for volume from the units and dimensions table in the Units section of Lessons.

FORMULA:   V = l l l
You know three lengths. You are asked to find a volume. For this question you need the volume equation from the formula section of Units in Lessons.

SOLVE: The equation is already solved for V, the volume. Substitute the known values from GIVEN into the working equation, simplify, and do the math. Here the math includes some DA to get the units right.

WHAT YOU WRITE DOWN SHOULD LOOK LIKE THIS:

GIVEN:   1 = 10.0 ft,     2 = 13.0 ft,     3 = 8.00 ft,

FIND:   a volume, V

By calculator it is:   10 x 13 x 8 x 12 x 12 x 12 x 2.54 x 2.54 x 2.54 ÷ 1000 =
Likely, you know some ways to make the math shorter.

ANSWER: Calculate the numerical answer, show your cancellation of the units, and show the raw answer (the units that remain and the number on your calculator). Change the raw answer to the correct number of significant digits and change into scientific notation if the answer is greater than 1000 or less than 1/1000.

PROBLEM #6, SET 2 - NUMBERS

6. A box of qrtpsdfgh measures 24.0 x 4.00 x 12.0 inches. It has a density of 38.0 pounds per cubic foot. What is the weight of qrtpsdfgh in the box? (Notice the English system density!)

This problem is likely best done with W5P method because it asks for a weight when you are given three lengths and a weight density. The units are all English system.

GIVEN:   l1 = 24.0 in,     l2 = 4.00 in,     l3 = 12.0 in,     D = 38.0 #/cu.ft.
You know this from the units and dimensions table in the Units section of Lessons.

FIND:   a weight, Fw
The question asks for a weight, Fw. You know the symbol for weight from the units and dimensions table in the Units section of Lessons.

FORMULAS:   D = Fw/V     V = l l l
You know three lengths and a weight density. You are asked to find a weight. The formulas come from the formula section of Units in Lessons.

SOLVE: The equation is already solved for V, the volume. Substitute the known values from GIVEN into the working equation, simplify, and do the math. Here the math includes some DA to get the units right.

WHAT YOU WRITE DOWN SHOULD LOOK LIKE THIS:

GIVEN:   1 = 24.0 in,     2 = 4.00 in,     3 = 12.0 in,     D = 38.0 #/cu.ft.

FIND:   a weight, Fw

The calculator math is:   38 x 24 x 4 x 12 ÷ 12 ÷ 12÷ 12 =   or just:   38 x 24 x 4 ÷ 12÷ 12 =

ANSWER: Calculate the numerical answer, show your cancellation of the units, and show the raw answer (the units that remain and the number on your calculator). Change the raw answer to the correct number of significant digits and change into scientific notation if the answer is greater than 1000 or less than 1/1000.

PROBLEM #7, SET 2 - NUMBERS

7. My basement was flooded after a big rain. The basement is rectangular 12.0 ft by 17.0 ft. The water was measured as it drained out at 51.0 cubic feet. How deep was the water in the basement?

This problem is likely best done with W5P method because it asks for a distance when you are given two lengths and a volume. The units are all English system.

GIVEN:   1 = 12.0 ft,     2 = 17.0 ft,     V = 51.0 cu.ft.
You know this from the units and dimensions table in the Units section of Lessons.

FIND:   a length, l
The question asks for a length, l. You know the symbol for length from the units and dimensions table in the Units section of Lessons.

FORMULA:   V = l l l
You know two lengths and a volume. You are asked to find a length. The formulas come from the formula section of Units in Lessons.

SOLVE: Solve for one of the lengths. Substitute the known values from GIVEN into the working equation, simplify, and do the math. Here the math includes some DA to get the units right.

WHAT YOU WRITE DOWN SHOULD LOOK LIKE THIS:

GIVEN:   1 = 12.0 ft,     2 = 17.0 ft,     V = 51.0 cu.ft.

FIND:   a length, l

The calculator math is:   51 ÷ 12 ÷ 17 x 12 =   or just:   51 ÷ 17 =

ANSWER: Calculate the numerical answer, show your cancellation of the units, and show the raw answer (the units that remain and the number on your calculator). Change the raw answer to the correct number of significant digits and change into scientific notation if the answer is greater than 1000 or less than 1/1000. No change from the raw answer is needed here.

PROBLEM #8, SET 2 - NUMBERS

8. A standard expressway lane is 11.0 ft wide. How long would a four lane expressway be if it had a square mile of pavement?

This problem is likely best done with W5P method because it asks for a distance when you are given one length and an area. The units are all English system.

GIVEN:   1 = 44.0 ft(4 lanes at 11 ft each!),     = 1 sq. mi.
You know this from the units and dimensions table in the Units section of Lessons.

FIND:   a length, l
The question asks for a length, l. You know the symbol for length from the units and dimensions table in the Units section of Lessons.

FORMULA:  A = l l
You know one length and an area. You are asked to find a length. The formulas come from the formula section of Units in Lessons.

SOLVE: Solve for one of the lengths. Substitute the known values from GIVEN into the working equation, simplify, and do the math. Here the math includes some DA to get the units right.

WHAT YOU WRITE DOWN SHOULD LOOK LIKE THIS:

GIVEN:   1 = 44.0 ft,     A = 1 sq. mi.

FIND:   a length, l

The calculator math is:   5280 ÷ 44 =

ANSWER: Calculate the numerical answer, show your cancellation of the units, and show the raw answer (the units that remain and the number on your calculator). Change the raw answer to the correct number of significant digits and change into scientific notation if the answer is greater than 1000 or less than 1/1000. The raw answer needs no change to it.

PROBLEM #9, SET 2 - NUMBERS

9. The density of gasoline is 720 g per liter. What is the mass of a jar of gasoline three cm by four cm by five cm?

This problem is likely best done with W5P method because it asks for a density when you are given three lengths and a mass. The units are all metric.

GIVEN:   l1 = 3 cm,   l2 = 4 cm,   l3 = 5 cm,   D = 720 g/L
You know this from the units and dimensions table in the Units section of Lessons.

FIND:   mass, m
The question asks for a mass, m. You know the symbol for length from the units and dimensions table in the Units section of Lessons.

FORMULA(S):  V = l l l     D = m/V
You know three lengths and a density. You are asked to find a mass. The formulas come from the formula section of Units in Lessons.

SOLVE: Solve for the mass. Substitute the known values from GIVEN into the working equation, simplify, and do the math. Here the math includes some DA to get the units right.

WHAT YOU WRITE DOWN SHOULD LOOK LIKE THIS:

GIVEN: 1 = 3 cm,   2 = 4 cm,   3 = 5 cm,   D = 720 g/L

FIND:   a length, l

The calculator math is:   720 x 3 x 4 x 5 ÷ 1000 =

ANSWER: Calculate the numerical answer, show your cancellation of the units, and show the raw answer (the units that remain and the number on your calculator). Change the raw answer to the correct number of significant digits and change into scientific notation if the answer is greater than 1000 or less than 1/1000. Once again, the raw answer does not need to be changed.

PROBLEM #10, SET 2 - NUMBERS

10. How many liters of water could you fit into a hot tub that measures 1.50m deep and three meters in diameter? How many people?
(1 cubic meter = 1000L)

This problem is likely best done with W5P method because it asks for the volume of a cylinder when you are given a length as height and another length as diameter. The units are all metric.

GIVEN:   = 11.0 ft,     d = 3 m (so the radius, r = 1.5 m)
You know this from the units and dimensions table in the Units section of Lessons.

FIND:   a volume, V
The question asks for a length, l. You know the symbol for volume from the units and dimensions table in the Units section of Lessons.

FORMULA:   Vc = pi r2
You know the radius and height of a cylinder. You are asked to find the volume. The formula comes from the formula section of Units in Lessons.

SOLVE: The volume is already solved for in the memorized equation. GIVEN into equation and do the math. Here the math includes some DA to get the units right.

WHAT YOU WRITE DOWN SHOULD LOOK LIKE THIS:

GIVEN:   = 11.0 ft,     d = 3 m (so the radius, r = 1.5 m)

FIND:   a volume, V

The calculator math is:   pi x 1.5 x 1.5 x 1.5 x 1000 =

ANSWER: Calculate the numerical answer, show your cancellation of the units, and show the raw answer (the units that remain and the number on your calculator). Change the raw answer to the correct number of significant digits and change into scientific notation if the answer is greater than 1000 or less than 1/1000. You can leave the raw answer as it is.

The answer to the second question is:   THE MORE, THE MERRIER.

PROBLEM #11, SET 2 - NUMBERS

11. Mr. Richland has a swimming pool 10.0 ft by 20.0 ft. He had a wild party and they threw his son's Volkswagen into the pool. The VW has a volume of 10 cubic feet. It was sealed up, but it sank because of all the lead bricks in it. (?!?!?*?*?)
(a) What is the area of the pool? (b) How much did the VW raise the water level?

A major idea here is that the volume of the VW is equal to the volume of displaced water. The displaced water can be measured as a layer on top of the other water in the pool.

Part (a) of the problem is easy and intuitive. Given two distances, what is the area?
Given = 10 ft and = 20 ft, = = (10 ft)(20 ft) = 200 ft2

The second part of this problem is likely best done with W5P method because it asks for a distance when you are given two lengths and a volume. The units are all English system.

GIVEN:   1 = 10.0 ft,     2 = 20.0 ft,     V = 10 cu. ft.
You know this from the units and dimensions table in the Units section of Lessons.

FIND:   a length, l
The question asks for a length, l. You know the symbol for length from the units and dimensions table in the Units section of Lessons.

FORMULA:  V = l l l
You know two lengths and a volume. You are asked to find a length. The formulas come from the formula section of Units in Lessons.

SOLVE: Solve for one of the lengths. Substitute the known values from GIVEN into the working equation, simplify, and do the math. Here the math includes some DA to get the units right.

WHAT YOU WRITE DOWN SHOULD LOOK LIKE THIS:

GIVEN:   1 = 10.0 ft,     2 = 20.0 ft,     V = 10 cu. ft.

FIND:   a length, l

The calculator math is:   10 ÷ 10 ÷ 20 x 12 =   or just:   12 ÷ 20 =

ANSWER: Calculate the numerical answer, show your cancellation of the units, and show the raw answer (the units that remain and the number on your calculator). Change the raw answer to the correct number of significant digits and change into scientific notation if the answer is greater than 1000 or less than 1/1000.

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