#19-12: during the rainy season, 4/9 of a foot of rain was collected in 2/3 of a day. at that rate, how much rain would be collected in one day?

2/3 ft rain

Step-by-step explanation:

You could setup a proportion:

2/3 day gives you 4/9 ft of rain

1     day gives you    x  ft of rain

This information is organized days in 1 column with rain fall in another column.  Even the rows are organized.  The 4/9 rain goes with the 2/3 day and the x rain goes with the 1 day.

I organized is so it is easy to setup the proportion:

Cross multiply:

Simplifying right hand side:

To solve this equation we can multiply the reciprocal of the coefficient of x on both sides to get x by itself:

The reason we do that is because when you multiply reciprocals you get 1. We wanted to obtain x by itself.  We got 1x which is still just x:

Now we can put this last part in your calculator.

You could multiply the fractions by multiplying straight across on top and bottom. Then reduce.

You could actually reduce then multiply.

I'm going to do the last option.

3/9=1/3 and 4/2=2 so we have:

There are other ways people like to think about this.

Maybe you are the type who likes this:

2/3 day ->4/9 ft rain

The remaining part of the day is 1/3.

Half of 2/3 is 1/3 since 1/3+1/3=2/3.

So if I half both sides I get:

1/3 day=2/9 ft rain

So (2/3+1/3) day=1 day implies we have (4/9+2/9) ft rain.

Simplifying:

1 day=6/9 ft rain

Simplifying the 6/9 by dividing top and bottom by 3 gives you:

1 day=2/3 ft rain

You could also think about units rate here because they are asking for one since they want to know what happens in 1 day.

So you do (4/9) divided by (2/3).

You can change division to multiplication by flipping the 2nd number (the one after the division sign):

(4/9) divided by (2/3)

(4/9)  times         (3/2)

12/18

12 and 18 have a common factor of 6 so divide top and bottom by 6:

2/3

So this unit rate says you get 2/3 ft rain per day.

We are looking for the rainfall for one day which would be 2/3 ft here.

a) $ 9.50

b) 17

c) y-intercept (0, 12). this is the initial amount of money that joe had prior to playing any game.

d) x-intercept ( 24, 0). this is the number of games that joe will play before he depletes the initial amount of money that he had.

e) slope is -0.5. joe's balance will reduce by 0.5 for every game that he plays.

f)

domain; [0,24]

range; [0,12]

g) x + 2y = 24

step-by-step explanation:

a)

if joe plays 5 games, this will imply that x = 5. substitute x = 5 in the equation and solve for y;

y = -.50x + 12

y = -0.50(5) + 12

y = 9.50

b)

if joe leaves the arcade with $3.50, this will mean that y = 3.50. plug in y = 3.50 in the given equation and solve for x;

y = -.50x + 12

3.50 = -0.50x + 12

0.50x = 12 - 3.50

0.5x = 8.5

x = 17

c)

the function given is   y = -.50x + 12.

the y-intercept is the point where the graph of this function will intersect the y-axis. at this point, the value of x is usually 0. plug in x = 0 in the given equation and determine y;

y = -0.50(0) + 12

y = 12

the y-intercept is thus (0, 12). this is the initial amount of money that joe had prior to playing any game.

d)

the x-intercept refers to the point where the graph of a function intersects with the x-axis. at this point, the value of y is 0. substitute y = 0 in the given equation and solve for x;

0 =   -.50x + 12

0.5x = 12

x = 24

x-intercept ( 24, 0). this is the number of games that joe will play before he depletes the initial amount of money that he had.

e)

the slope of the given equation is -0.5. the equation has been given in slope-intercept form. thus the slope is simply the coefficient of x.

joe's balance will reduce by 0.5 for every game that he plays. this is the interpretation of the slope.

f)

domain is the set of x values for which the function is real and defined. the domain will thus be; [0,24]. this will restrict the balance to only assume positive values.

on the other hand, the range is the set of y values for which a function is real and defined. the range in our case will be [0,12]. joe can have between $0 and $12 balance.

g)

the standard form of the equation of a straight line is;

ax + by = c

where a, b, and c are integers.

our equation is given as;

y = -.50x + 12

re-writing the equation we have;

0.5x + y = 12

divide all through by 0.5;

x + 2y = 24

the answer is b.

step-by-step explanation:

by the trigonometric ratio tangent, tangent equals opposite divided by adjacent.

therefore, tan(50) equals ac/3

multiply each side by 3 to get: ac=3tan(50)

the answer is b.