math geniuses!! pls help me!!! my teacher doesn't even understand this problem?

Question

A rainstorm lasts for 4.5 hours, building from a drizzle to a heavy rain, then dropping back to a drizzle. the rate, r, of the rain (in inches per hour) is given by

r = 0.5t ; 0 (less than or equaly symbol) t (less than or equaly symbol) 2

r = 1.8 - 0.4t ; 2 < t (less than or equaly symbol) 4.5

where t is the time in hours, what is the total rainfall?

Answer 1

Some responders are giving the right answer, but they're all making it too complicated.
You can do it with calculus, but there's no reason to make it that hard.

During the first two hours, the rainfall increases linearly from 0 to 1 inch per hour. So it averages 0.5 inches per hour during those 2 hours.

During the last 2.5 hours, the rainfall decreases linearly from 1 to 0 inches per hour. So it averages 0.5 inches per hour during those 2.5 hours.

Bottom line: Over the entire 4.5 hours, the average rate was 0.5 inches per hour, so the total rainfall is
4.5 hours x 0.5 inches/hour = 2.25 inches of rain.

Answer 2

r={t/2 where 0≤t≤2
9/5-2t/5 where 2
You are given the rate at which rain falls per hour, and asked to find the total rainfall. This will be the integral of the rate of rainfall over the interval [0, 9/2]. So:

[0, 9/2]∫r dt

Breaking this up into two subintervals:

[0, 2]∫t/2 dt + [2, 9/2]∫9/5-2t/5 dt

Integrating:

t²/4 |[0, 2] + (9t/5 - t²/5)|[2, 9/2]

Evaluating at the endpoints:

(4/4 - 0) + ((81/10 - 81/20) - (18/5 - 4/5))
1 + 81/20 - 14/5
20/20 + 81/20 - 56/20
45/20
9/4
2.25

Edit: Maths Rocks, 8.1- 4.05 - 3.6 +0.8 = 1.25, not 1.245.

Answer 3

There are two triangles: one that is 2 by 1 and one that is 2.5 by 1. Triangles have area = 1/2 base times height. Both these triangles are one unit tall.

They kind of look like this: /\, but flatter.

0.5*(2*1) + 0.5*(2.5*1) = 2.25 inches

It's NOT a coincidence that this is the same area as the area of a triangle with height 1 and base 4.5.

There's no need for calculus here.

Answer 4

a million,. what number diverse 5-digit numbers could properly be formed utilizing the digits a million; 2; 3; 4; and 5; without repetition? (you do not could record a number of those numbers.) there are 5 numbers to elect from. & you'll use all of them. the answer is 5!=120 2,. If I particularly have 3 diverse pairs of boots, 2 outfits, and a couple of jackets, what number diverse clothing are accessible to placed on? 3*2*2=12 3,. We toss a coin two times in a row. record all accessible consequences. (as an party, one accessible effect is HT, i.e. head for the first, tail for the 2d toss.) HH HT TH TT

Answer 5

T = 4.5
r = .5t" for t <= 2
r =1.8 =.4t for t > 2

so for hours 0 -2 rainfall = .5 * 2 = 1"
4.5 - 2 = 2.5 hours remaining

r = 1.8 -.4 *2.5
r = 1.8 - 1 = .8"

so total rainfall = 1.8" total rainfall

Answer 6

for the first 2 hours:
r = (.5 inches/hour)(2hours)= 1 inch
for the next 2.5 hours:
r = (1.8 - .4(2.5)) = .8 inch

so, the total rainfall is r = 1 +.8 = 1.8 inches

Answer 7

i think u integrate both expressions
R=rain in inches
dR/dt=r then:
dR/dt=0.5t
and
dR/dt-1.8-0.4t
integrate first expression:
R= [0.5t^2/2] limit 0-2
= 0.25t^2]0-2
= 0.25(2)^2 -(0.25(0)^2
=1 inch
integrate second expression
R= [1.8t- 0.4t^2/2]limit 2-4.5
R=1.8t-0.2t^2]2-4.5
R= [1.8(4.5) -0.2(4.5)^2]
- [ 1.8(2)-0.2(2)^2]
=8.1- 4.05 - 3.6 +0.8
=1.245inches

Total rainfall =2.245inches
=