Transcript Quiz 1-3

Quiz 1-3
1. Solve for x:
4x
 3  10
5
2. Solve for x:
2( x  3)  2(2 x  1)
3. Simplify:
2 x  3x  4 x  2 x
4
2
4
4. What property is illustrated below?
2  3  5  5  (3  2)
Quiz 1-5
Time (min)
1. Height (ft)
0
1
36,000 32,800
2
3
4
29,600
26,400
23,200
The able shows the altitude of an airplane. What is its
altitude at the 8 minute point?
2. Wood shop: A piece of wood is 72” long. You cut
the wood into 3 pieces. The 2nd piece is 6” longer
than the 1st piece. The 3rd piece is 6” longer than the
than the 2nd piece.
a. Draw a diagram showing relative lengths of the
pieces.
b. Write an equation showing the length of each
piece (use only one variable: x = length of 1st piece.
c. How long is the 1st piece?
1-5
Time (min)
1. Height (ft)
Quiz
0
1
36,000 32,800
2
3
4
29,600
26,400
23,200
The table shows the altitude of an airplane.
What is its altitude at the 8 minute point?
2. A car is traveling at 88 feet per second. How long
will it take to travel 120 miles?
Quiz
1-5
1. A salesperson has a base salary of $20,000 per year.
She earns a commission based upon her total sales.
Her commission is 10% of her total sales. If her total
annual income was $55,000, what was her total sales?
2. You leave Roy and travel south on the freeway at 65 mph.
At the same time your friend travels north from Roy at a
speed of 50 mph. How long would it take for you to be 200
miles apart?
No quiz today
What problem from the homework do
you want me to work?
Finish
Section 1-5
Time-Distance word problems
Speed/Distance Model: (involves time)
Distance = (speed) (time)
d = r*t
Speed is a “rate”
(distance per unit time)
This is a ‘gotcha.’ All units (hours, minutes, feet, miles,
etc., MUST be CONSISTENT throughout the problem!!!!
Speed/Distance Model: (involves time)
Distance = (speed) (time)
Example:
d = r*t
It takes you 5 hours to drive to St. George. St. George is 300
miles away. How fast were you going?
1. Write the formula
d = r*t
2. Identify the quantities from the formula that are given in
the problem: d = ?, r = ?, t = ?
d = 300 miles, r = ?, t = 5 hours
Speed/Distance Model: (involves time)
Distance = (speed) (time)
Example:
d = r*t
It takes you 5 hours to drive to St. George. St. George is 300
miles away. How fast were you going?
3. Replace the values given into the formula.
300 miles = r * 5 hours
4. Solve for the unknown variable.
300 miles = r * 5 hours
÷ 5 hours
÷ 5 hours
300 miles
r
5hours
miles
 60
hour
Speed/Distance Model: (involves time)
Distance = (speed) (time)
Example:
d = r*t
A plane flew at a speed of 300 miles/hr for 7 hours. How far did
it fly?
1. Write the formula
d = r*t
2. Identify the quantities from the formula that are given in
the problem: d = ?, r = ?, t = ?
d = ?, r = 300 miles/hr, t = 7 hours
Speed/Distance Model: (involves time)
Distance = (speed) (time)
Example:
d = r*t
A plane flew at a speed of 300 miles/hr for 7 hours. How far did
it fly?
d = r*t
1. Write the formula
2. Identify the quantities from the formula that are given in
the problem: d = ?, r = ?, t = ?
d = ?, r = 300 miles/hr, t = 7 hours
3. Replace the values given into the formula.
d= 300 miles/hr * 7 hours
4. Solve for the unknown variable.
300 miles
d
* 7 hours
hour
300 miles 7 hours
d
*
hour
1
d  2100 .miles
Speed/Distance Model: (involves time)
Distance = (speed) (time)
d = r*t
Example:
A plane flew 4000 miles in 7 hours. What was its speed?
3. Replace the values given into the formula.
4000 = x miles/hr * 7 hours
4. Solve for the unknown variable.
1
* 4000 miles  x miles * 7 hours * 1
7 hours
hour
1
7 hours
4000 miles x miles

7 hrs
hour
571.4 miles
 speed
hr
Your turn ½ point of the equation, ½ point for the solution.
Distance = (speed) (time)
d = r*t
1. What would the speed have to be to travel 1000 miles in
6 hours?
2. How long would it take to travel 1500 miles if your
speed was 200 miles per hour?
Two people traveling
(1) Same direction
d = r*t
You leave Roy and travel south at 65 mph. Your
friend leaves 2 hours later. How long would she have to
travel to catch up to you if she is going 75 mph?
distanceyours  rateyours * timeyours
distancehers  ratehers * timehers
Who traveled further, you or your friend?
distanceyours  distancehers
Two people traveling
d = r*t
(1) Same direction
You leave Roy and travel south at 65 mph. Your
friend leaves 2 hours later. How long would she have to
travel to catch up to you if she is going 75 mph?
distanceyours  distancehers
distanceyours  rateyours * timeyours  distancehers  ratehers * timehers
rateyours * timeyours  ratehers * timehers
How do we relate her travel time to your travel time?
timeyours  timehers  2hrs
rateyours * (timehers  2hrs)  ratehers * timehers
65miles
75miles
* (t  2hrs ) 
*t
hr
hr
65(t  2)  75t
65t  130  75t
130  15t
t  8.7 hrs
Summary
(1) Same direction
d = r*t
distanceyours  distancehers
rateyours * timeyours  ratehers * timehers
Replace the variables with numbers from the problem.
If there are 2 unknown variables, you need to find a way
to relate the two. For example:
timeyours  timehers  2hrs
When you have only one variable, you can solve for it.
Your turn
(1) Same direction
d = r*t
3. Your friend travels north at 20 mph for an hour, then you
follow at 30 mph. How long will it take to catch up?
4. Your friend travels east at 50 mph for 3 hours, then you
follow. It takes you 5 hours to catch up. How fast were you
going?
Two people traveling
(2) Opposite direction
d = r*t
You leave Roy and travel south at 65 mph. Your
friend travels north at 50 mph. How long will it be until you are
350 miles apart?
distancehers  ratehers * timehers
distanceyours  rateyours * timeyours
350 miles
Total distance problem. distanceyours  distancehers  total distance
rateyours * timeyours  ratehers * timehers  350
65* timeyours  50* timehers  350
How do we relate her travel time to your travel time?
timeyours  timehers
65t  50t  350
115 t  350
t  3.04 hrs
Summary
(2) Opposite direction
d = r*t
distanceyours  distancehers  total distance
rate
yours
* timeyours  ratehers * timehers   total distance
Replace the variables with numbers from the problem.
If there are 2 unknown variables, you need to find a way
to relate the two. For example:
timeyours  timehers  2hrs
When you have only one variable, you can solve for it.
Two people traveling
(2) Opposite direction
d = r*t
5. You and your friend both leave Roy at the same time in
opposite directions. Your speed is 30 mph and his speed
is 55 mph. How long will it be until you are 280 miles apart?
6. You and your friend both leave Roy at the same time in
opposite directions. Your speed is 50 mph. After 8 hours
you are 600 miles apart. What was his speed?
Homework:
Section 1 – 5
Finish the time-distance problems from the assignment
work sheet.
Section 1-4
Rewrite Formulas and Equations.
Homework:
Section 1 – 4
Problems (evens):
2-16 (for problems 8-14 these are two part problems; 1st
solve for the indicated variable then plug in a value)
22-28: book says solve and plug in like 2-16 above but I
just want you to solve for the indicated variable (don’t
plug in).
40-52
(19 problems)
Vocabulary
Solve the single variable equation: Use properties of
equality to rewrite the equation as an equivalent equation
with the variable on one side of the equal sign and a
number on the other side.
Solve for a variable (more then one variable in the equation):
Use properties of equality to rewrite the equation as an
equivalent equation with the specified variable on one side of
the equal sign and all other terms on the other side.
Solve for “x”
x + 1 = 5
-1
-1
x
=
4
Solve for the variable: Use properties of
equality to rewrite the equation as an
equivalent equation with the variable on
one side of the equal sign and a number
on the other side.
Solve for ‘x’
4 + 2x + y = 6
-4
-4
2x + y = 2
-y
2x
÷2
-y
= 2–y
÷2
2 y
x
2
Solve for the variable: Use properties of equality to rewrite
the equation as an equivalent equation with the specified
variable on one side of the equal sign and all other terms
on the other side.
Solve for “x”
yx – 2
= 4
+2
+2
yx
= 6
÷y
÷y
6
x
y
Solve for the variable: Use properties of equality to rewrite
the equation as an equivalent equation with the specified
variable on one side of the equal sign and all other terms
on the other side.
Your turn:
7. Solve for ‘k’
2k  3m  5
8. Solve for ‘k’
4m  3ky  7
9. Solve for ‘k’
7k  3
 4x
2y
Vocabulary
Quantity: An measure of a real world physical property
(length, width, temperature, pressure, weight, mass, etc.).
Formula: An equation that relates two or more quantities,
usually represented by variables.
Areacircle  r
2
Arearectangle  l * w
Perimeterrectangle  ?
Prectangle  2L  2w
L
W
W
L
Formulas are used extensively in
science.
Science and math come together when mathematical
equations are used to describe the physical world.
Once a formula is known then scientists can use the
equation to predict the value of unknown variables in the
formula.
Circumference
C = πd
d = 2r
What real world quantity
does “d” represent?
What real world quantity
does “r” represent?
C = π*2r
Since d = 2r, we can replace ‘d’ in the
circumference formula with ‘2r’.
C = 2πr
What property allows us to re-write
the formula like this ?
Solve for radius
C  2r
÷ 2π
÷ 2π
In this form, we say that
‘c’ is a function of ‘r’.
We will now solve for “r”
C
r
2
C
r
2
In this form, we say that
‘r’ is a function of ‘c’.
Circumference
Your turn: for the area of a triangle formula:
1
A  bh
2
10. Solve for “b”
(This is ‘A’ is a function of ‘b’ and ‘h’.)
We call this new version of the formula
“b” is a function of “h” and “A”
11. Solve for “h”.
12. What do you call this new version of the formula?
Your Turn:
1
A  b1  b2 h
2
13. Solve for ‘h’.
(Area of a trapezoid: where the length
of the parallel bases are b1..and..b2 
and the distance between them is ‘h’.)
b1
h
14. Solve for b2
b2
What if two terms have the variable you’re trying
to solve for?
Solve for ‘x’.
‘x’ is common to both terms 
factor it out (reverse distributive
property).
How do you turn (3y – 2) into
a “one” so that it disappears
on the left side of the equation?
3xy  2 x  10
x(3 y  2)  10
÷(3y – 2)
÷(3y – 2)
10
x
(3 y  2)
Example
Solve the equation for “y”.
9y + 6xy = 30
Use “reverse distributive property
9 y  6 xy  30
What is “common” to both of
the left side terms?
y(9  6 x)  30
÷(9 + 6x)
“Factor out” the common term
÷ (9 + 6x)
30
y
(9  6 x)
“same thing left/right”
Your turn:
15. Solve for ‘x’. xy  3x  40
16. In problem #15, if y = 5, x = ?
17. Solve for ‘y’. xy  2 yx  5 y  1
Solving formula Problems
The perimeter of a rectangular back yard is 41 feet. Its
length is 12 feet. What is its width?
Draw the picture
Write the formula
Perimeterrectangle  ?
P  41 ft
Prectangle  2L  2w
Replace known variables in
the formula with constants
41ft  2(12 ft) 2w
41  24  2w
-24
-24
17 
2w
Solve for the
variable
W
12 ft
17  2w
÷2
÷2
17
ft  w  8.5 ft
2
Solving formula Problems
1. Draw the picture (it helps to see it)
2. Write the formula
3. Replace known variables in the formula with constants
4. Solve for the variable
1
Prectangle  2L  2w Areatriangle  1 2 bh Atrapezoid  b1  b2 h
2
Your turn:
18. If the base of a triangle is 4 inches and its area is
15 square inches, what is its height?
19. The area of a trapezoid is 40 square feet. The length of
one base is 8 feet and its height is 3 feet, what is the
length of the other base?
20. The perimeter of a rectangle is 100 miles. It is 22 miles
long. How wide is the rectangle?
End here
Using formulas so solve real world problems.
What is the profit model (words describing the relationship
between profit, $ from sales, and $ going for expenses.) ?
P  $ from..sales...subtract...costs.
(From a previous example) the profit for selling ‘c’ candles at
$3 each when the cost to rent the booth and buy supplies is $120.
P = 3c - 120
How many candles must be sold to
have a profit of $500 ?
Using formulas so solve real world problems.
How many candles must be sold to have a profit of $525 ?
P = 3c - 120
Method 1
1. Solve for the variable
P  3c  120
+120
+120
P  120  3c
÷3
÷3
P  120
c
3
2. Plug numbers into the formula.
P  120
c
3
525  120
c
3
645
c
3
c  215
Using formulas so solve real world problems.
P = 3c - 120
How many candles must be sold to have a profit of $525 ?
Method 2
1. Plug numbers into the formula.
P  3c  120
525  3c  120
2. Solve for the variable
525  3c  120
+120
645  3c
÷3
215  c
Same answer
+120
÷3