November 03, 2014
Precalc Warm Up # 10-1
groups
In my class of 36 students, kids are seated in rows such
that every row contains an equal number of people. But
if I increase the number of rows by 3, then there would
need to be 2 less kids in each row. How many rows are
there?
Let r = # rows originally and n = # people in each row.
r. n = 36
(r +3)(n - 2) = 36
r.n - 2r + 3n - 6 = 36
36 - 2r + 3n - 6 = 36
__ so
-2r + 3n = 6 and from line 1, n =36
r
__) = 6
-2r + 3( 36
2
r
-2r + 108
= 6r
0 = 2r 2 + 6r - 108
0 = r 2+ 3r - 54
0 = (r + 9)(r - 6)
Since we can't have - 9 rows, there must have been 6
November 03, 2014
Like 17 from last night's HW:
"Work" problems involve situations such as two people working
together to paint a house. You are usually told how long
each person takes to paint a similarly-sized house alone, and
you are asked how long it will take the two of them to paint the
house when they work together.
There is a "trick" to doing work problems: you have to think of
the problem in terms of how much each person / machine /
whatever does in a given unit of time. For instance:
November 03, 2014
Suppose one painter can paint the entire house in
twelve hours, and the second painter takes eight
hours. How long would it take the two painters
together to paint the house?
If the first painter can do the entire job in twelve hours
and the second painter can do it in eight hours, then (this
here is the trick!) the first guy can do 1/12 of the job per
hour, and the second guy can do 1/8 per hour.
If they work on it for r hours together, what fraction of
the job does the first painter do?
What fraction of the job does the 2nd painter do?
What should these fractions add up to?
November 03, 2014
November 03, 2014
Suppose one painter can paint the entire house in
twelve hours, and the second painter takes eight
hours. How long would it take the two painters
together to paint the house?
Let r = # hours working together.
Painter A does
Painter B does
1
___
12
1
___
8
1
___
per hour , so the fraction she gets done is 12 r
1 r
per hour, so the fraction he gets done is ___
8
Together these fractions add up to 1,
1
___
so
12
1
r + ___
8 r = 1
2r + 3r = 24
5r = 24
24
r = ___
5
24
___
It takes them 5 of an hour working together, or 4.8
hours, which is 4 hours 48 minutes
November 03, 2014
Suppose one painter can paint the entire house in
twelve hours, and the second painter takes eight
hours. How long would it take the two painters
together to paint the house?
A common
mistake is to
AVERAGE their
times. Why does
this not make
sense?
November 03, 2014
November 03, 2014
November 03, 2014
November 03, 2014
Going over 17 from last night:
Two taps having different rates of flow are used to fill a
large water tank. If tap A is used on its own it will take 5
hours longer to fill the tank than it would tap B to fill it on its
own. Together, the taps would fill the tank in 6 hours.
a. how long for tap A to fill it?
1
__
Let A = # hrs for tap A alone to fill it. Then it gets A
done
per hour
1
Then A - 5 = # hrs for tap B to fill it alone, so it gets __
A-5
done per hour.
1
1 .6
__
__ .
6+
=1
A
A-5
6(A - 5) + 6A = A (A - 5)
6A - 30 + 6A = A2 - 5A
0 = A2 - 17A + 30
0 = (A - 15)(A - 2)
Since 2 hours doesn't make sense (why?), it takes tank A
15 hours to fill it alone
b. How long for tap B to fill it?
It takes tank B 10 hours alone to fill it
November 03, 2014
Going over 18 from last night
2 chair manufacturers have been in competition for
many years. Chair One had to fulfill an order for 810
chairs within a set time, while Chair Two had to fulfill an
order for 900 chairs in the same amount of time.
Competition led these companies to complete the task 3
days and 6 days in advance respectively. We know
Chair One makes 4 less chairs a day than Chair Two.
How many chairs does each make per day?
Let x = # chairs that Chair One normally makes per day,
and let d = # days normally
x(d - 3) = 810
xd - 3x = 810
xd = 3x + 810
Then x + 4 = # chairs that Chair Two normally makes a
day with d = #days normally
(x+4)(d - 6) = 900 xd - 6x + 4d - 24 = 900
xd = 6x - 4d + 924
3x + 810 = 6x - 4d + 924
-114 = 3x - 4d
810
and from the green, x = ____
d-3
810
-114 = 3 · ____
d - 3 - 4d
-114(d-3) = 2430 - 4d(d-3)
-114d + 342 = 2430 - 4d2 + 12d
4d2 - 126d - 2088 = 0
using the quadratic formula , d = -12 or 43.5
43.5 days
x (43.5 - 3) = 810
x (40.5) = 810
x = 20 chairs for Chair One
and 24 chairs for Chair Two.
November 03, 2014
Any other questions from #9-5?
November 03, 2014
November 03, 2014
November 03, 2014
November 03, 2014
In groups:
Lucy and Talia go trick or treating. In years past, Lucy has filled her bag
of candy in 2 hours, and Talia has filled hers in 3. This year they plan to
work together to fill a bag. How long will it take them working together?
Let t=time in hours it takes them to fill a bag working together.
1 bag per hour and
Lucy's rate is __
21
Talia's rate is __
bag per hour,
3
and they work for t hours, so
__
1
2 t
1
+ __
t =1
3
3t + 2t = 6
5t
=6
t
6 of an hour, so 1 hour and 12 minutes
= __
5
November 03, 2014
#10-1 p. 63
1,4,10, 12,13,15,16,19
The test is moved to Thursday.