Variable Relationships - Algebra 2

Find the pattern of the sequence:

This pattern is so the next number in the sequence would be

Find the pattern in this sequence of numbers:

In this case, the pattern is adding 13n to the previous number where n= how many numbers came before the current number.

so the first number we are looking for would be:

the second number we are looking for would be:

To extrapolate the results of the study out to 53 minutes, first we have to determine an equation representing the relationship between time passed and amount of water; we can write our equation in slope-intercept form:

Where our y-axis represents amount of water and the x-axis represents time. We can pick 2 points and label them Point 1 and Point 2; looking back at the table:

We label Point 1 as and Point 2 as ; we plug these points into the slope formula as follows:

So, the slope of our line that describes how much water is in the boat is ; to find our term, the y-intercept, we need to pick a point on the graph and plug in our slope to solve for y-intercept. Let's once again choose the point :

Simplify the expression and we find that b=0, so our slope-intercept equation is:

Plugging in a value of 53 for , we find that:

So the answer is 42.4 gallons of water in the boat after 53 minutes.

If varies directly with the square root of , then for some constant of variation ,

If , then ; therefore, the equation becomes

,

or

.

Divide by 5 to get , making the equation

.

If , then .

Since varies directly with , where is a constant.

1. Solve for when and .

2. Use your equation to solve for when .

Since varies indirectly with ,

1. Solve for when and .

2. Use the equation you found to solve for when .

1. Since varies directly with :

with K being some constant.

2. Solve for K using the x and y values given:

3. Use the equation you solved for to find the value of y:

1. Since varies indirectly with :

2. Use the given x and y values to determine the value of K:

3. Using the equation along with the value of K, find the value of y when x=5:

1. Since varies directly with :

2. Use the values given for x and y to solve for K:

3. Use your new equation with the K you solved for to solve for y when x=27:

1. Since y varies indirectly with :

2. Solve for K using the x and y values given:

3. Using the equation you created by solving for K, find y when x=100:

Using our two known points, we can use interpolation to determine the value at any point between them with the following formula:

Where is our first given point, is our second given point, and is the point we want to find. We know our two given points, as well as the x value of our unknown point, so now all we must do is plug in all of our known values and solve for y, our only unknown:

We want to do a linear interpolation since the relationship between workers and units can be assumed to be linear. This means there is a constant slope between the points, so the slope between two known points will be equal to the slope between the point we are trying to find and some known point. This is expressed in the relation:

,

where and are the points we want to find and and are known. We choose the known points to be those that are just to the left and right of the point we are trying to find,

and .

Plugging these into our interpolation formula and knowing , we can find , the units output per day.

.

Simplifying and rearranging to solve for :

.

So there are units produced when the number of workers is .

Use the formula for interpolation to determine the value of y:

We will use (30, 51) as our x2 and y2 and (20, 36) as our x1 and y1 and we will solve for y using 26.5 for x.

Use the formula for interpolation:

We will use (30, 15) as x2 and y2, (15, 10) as x1 and y1, and solve for y when x=17.9:

Using linear interpolation means that we draw a line between the points on our data set and use that line to estimate a value that lies between two data points; in this case, we have the data from Mary's 13th and 15th birthdays, so we can describe a line between those two points and estimate her height at 14. Our line will be written in slope-intercept form:

Where the variable represents Mary's Age in years and the variable represents her height in inches. First, we need to find the slope. Using 2 points on our table and as point 1 and point 2, respectively, we plug these values into our slope formula:

Next, we find the y-intercept by plugging in our slope (which we just found) and a point from our table (we'll stick with ) and solving for :

Simplify:

Subtract 35.75 from both sides to solve for :

The equation of our interpolation line is:

So, to get an estimate of Mary's height on her 14th birthday, we plug in and solve for :

Our estimate of Mary's height at is

Write the interpolation formula.

Identify and substitute the values.

Simplify the fraction.

The answer is:

Find the pattern of the sequence:

This pattern is so the next number in the sequence would be

Find the pattern in this sequence of numbers:

In this case, the pattern is adding 13n to the previous number where n= how many numbers came before the current number.

so the first number we are looking for would be:

the second number we are looking for would be:

To extrapolate the results of the study out to 53 minutes, first we have to determine an equation representing the relationship between time passed and amount of water; we can write our equation in slope-intercept form:

Where our y-axis represents amount of water and the x-axis represents time. We can pick 2 points and label them Point 1 and Point 2; looking back at the table:

We label Point 1 as and Point 2 as ; we plug these points into the slope formula as follows:

So, the slope of our line that describes how much water is in the boat is ; to find our term, the y-intercept, we need to pick a point on the graph and plug in our slope to solve for y-intercept. Let's once again choose the point :

Simplify the expression and we find that b=0, so our slope-intercept equation is:

Plugging in a value of 53 for , we find that:

So the answer is 42.4 gallons of water in the boat after 53 minutes.

If varies directly with the square root of , then for some constant of variation ,

If , then ; therefore, the equation becomes

,

or

.

Divide by 5 to get , making the equation

.

If , then .