1.) By increasing her usual speed by 20 kilometers per hour, a bus driver decreases the time on a 20-kilometer trip by 5 minutes. Find the usual speed.
2.) The height of an object thrown upward with an initial velocity of 64 feet per second is given by the formula h = −16t2 + 64t, where t is the time in seconds. How long will it take the object to reach a height of 64 feet?
3.) A jazz group on tour has been drawing average crowds of 500 people. It is projected that for every $1 increase in the $11 ticket price, the average attendance will decrease by 50. At what ticket price will nightly receipts be $4,900?
4.) Working together, Sarah and Heidi can clean the garage in 3 hours. If they work alone, it takes Heidi 8 hours longer than it takes Sarah. How long would it take Heidi to clean the garage alone?
Update:i got the answers to 2 and 4 so now i just need help with 1 and 3
Answers & Comments
Verified answer
1.) By increasing her usual speed by 20 kilometers per hour, a bus driver decreases the time on a 20-kilometer trip by 5 minutes. Find the usual speed.
Usual speed:
Distance = 20
Rate = 20/t
Time = t
Increased speed:
Distance = 20
Rate = 20/t + 20
Time = t - 1/12
t - 1/12 = 20/(20/t + 20)
(20/t + 20)(t - 1/12) = 20
20 - 20/12t + 20t - 20/12 = 20
-20/12t + 20t = 20/12
-20/12t + (20t)(12t)/12t = 20/12
(-20 + 240t²)/12t = 5/3
3(-20 + 240t²) = 60t
-60 + 720t² = 60t
720t² - 60t - 60 = 0
12t² - t - 1 = 0
Use the quadratic formula to find that t = 1/3 and t = -1/4. Since the time obviously cannot be a negative number, that means t = 1/3.
Usual speed:
Rate = 20/t = 20/(1/3) = 60
The usual speed is 60 kph.
-------------------------------------------------------------------------------------------------------
2.) The height of an object thrown upward with an initial velocity of 64 feet per second is given by the formula h = −16t2 + 64t, where t is the time in seconds. How long will it take the object to reach a height of 64 feet?
h = −16t2 + 64t
64 = -16t² + 64t
-16t² + 64t - 64 = 0
-t² + 4t - 4 = 0
Use the quadratic formula to find that t = 2.
The time it takes the object to reach 64 feet is 2 seconds.
-------------------------------------------------------------------------------------------------------
3.) A jazz group on tour has been drawing average crowds of 500 people. It is projected that for every $1 increase in the $11 ticket price, the average attendance will decrease by 50. At what ticket price will nightly receipts be $4,900?
Ticket price = 11 + p
(11 + p)(500 - 50p) = 4900
5500 - 550p + 500p - 50p² = 4900
-50p² - 50p + 600 = 0
-p² - p + 12 = 0
Use the quadratic formula to find p = 3 and p = -4. Since we are talking about INCREASING the price of the tickets only, we are left with p = 3.
Ticket price = 11 + p = 14
When the ticket price is $14, the nightly receipts will be $4900.
----------------------------------------------------------------------------------------------------------
4.) Working together, Sarah and Heidi can clean the garage in 3 hours. If they work alone, it takes Heidi 8 hours longer than it takes Sarah. How long would it take Heidi to clean the garage alone?
Sarah's rate + Heidi's rate = Rate working together
Let s = hours it takes Sarah to clean the garage working alone
Let s + 8 = hours it takes Heidi to clean the garage working alone
1/s + 1/(s+8) = 1/3
[3(s+8) / 3s(s+8)] + [3s / 3s(s+8)] = [s(s+8) / 3s(s+8)]
3(s+8) + 3s = s(s+8)
3s + 24 + 3s = s² + 8s
6s + 24 = s² + 8s
-s² - 2s + 24 = 0
Use the quadratic formula to find s = 4 and s = -6. Since Sarah cannot work at a negative rate, we are left with s = 4.
Let s + 8 = hours it takes Heidi to clean the garage working alone = 12
It takes Heidi 12 hours to clean the garage working alone.
============================================================
algebra the worst thing man has come up with !