Question 71862: My daughter brought this problem home. She seems lost and I have no clue how to help her. My math skills are really rusty it seems. Here's the problem:
In order to hunt hippopotamuses, a hunter must have a hippopotamus hunting license. Since the hunter can sell the hippos he catches, he can use the proceeds to pay for part or the entire license. If he catches only 3 hippos, he is still in debt by $2050. If he catches 7 hippos he makes a profit of $1550. The African Game and Wildlife Commission allows a limit of 10 hippos per hunter.
We're asked to define the independent and dependent variables. I figured that the profit is a dependent upon the number caught.
The cost of the license is independent - right?
Where I can't even get a clue is how to plot points on a straight line for this problem
and
how to find the y intercept and slope of the line.
please help.
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! Draw a set of coordinate axes. On the vertical axis label each major mark in $500 units
so that from the origin you can go up about 10 units (representing up to the value +$5000).
Also go down about 10 units (representing down to the value -$5000). On the horizontal
axis number from the origin in units of 1 ... and go out to 10 or 11 units.
.
Now we are good to go. The horizontal axis is the independent variable units and these
units represent the number of hippos ... from 0 to 10 hippos.
.
The vertical axis is the dependent variable dollars, and it is where you plot the loss or gain
depending on how many hippos you capture.
.
You are given two points on the graph ... you know that if you catch 3 hippos you are
still in debt by $2050 so you represent that as minus $2050 0r -$2050. On the x-axis start
at zero and count out +3 to the right. With your pencil at that point count vertically
down to the point where you are at -$2050. At $500 each unit you need to count down 4.1 units.
Stop. Mark that point. Call it (3, - 2050) because it represents 3 units along x and
2050 dollars below the break-even point.
.
Now do the same thing with the other point you are given. At 7 hippos you will be 7 units
from zero on the x-axis, and from that point you need to go vertically up to the representation
for $1550. Since you are above the x-axis the money is in the positive area and it
represents a gain.
.
When you have these two points plotted, put a straight-edge along the two points and
draw a straight line through the two points ... extending this line in both directions
beyond the two points.
.
Good enough. Now you have the graph of the line. The problem asks you to find the slope
of this line. Easy enough. Lightly draw a horizontal line through the point (3, -$2050)
and then lightly draw a vertical line down from the point (7, +$1550). Notice how these
two lines intersect to form a right angle. The slope of the graph is the ratio of the
lengths of two line segments ... it is the length of the line from the point (7, $1550) down
to the right angle intersection divided by the length of the line from the point (3, -$2550)
to the right angle intersection point. Let's find the length of the line down from the
point (7, +$1550) vertically down to the intersection point. At the top end of this line
the y-value is +$1550 and at the bottom end of this line the corresponding y-value
is - $2050. So the total length of the line is $1550 + $2050 = $3600. [You can also look
at this as subtracting -$2050 from +$1550 ... and [$1550 - (-$2050)] is also $3600.
.
Next let's find the length of the line from the point (3, -$2050) to the right angle
intersection. Can you see that at the left end (at the point 3,-$2050) the horizontal
value is 3 hippos? Then at the other end of this line (at the right angle intersection)
the horizontal value is 7. So this horizontal length runs from 3 to 7 hippos, and
therefore the length of the line is 4 ... (7 - 3 = 4).
.
Now you are ready to find the slope by dividing the rise ($3600) by the horizontal
run (+4) and when you do, the answer will be $900 per hippo. That means that for
every hippo captured you will gain $900. Remember this slope. Note that the
slope is
positive if the graph goes up as you move to the right. If the graph goes down as you
move to the right the slope is negative. In this problem, therefore, the slope is positive
... up and to the right.
.
You can write an equation for the line. One form of equation for a straight line is called
the slope intercept form. This form is:
.
y = mx + b
.
Remember that your dollar values are on the y-axis and your hippo values are on the x-axis.
In this equation form m is the slope and b is the point where the graph crosses the y-axis.
Let's put the slope into the equation:
.
y = 900x + b
.
We can find the value of b by substituting the values for one of the two given points
into this equation. Let's take the point (7, 1550). So in our equation we are going
to substitute 7 for x and 1550 for y. When we do the equation becomes:
.
1550 = (900*7) + b
.
Simplifying this by doing the multiplication results in:
.
1550 = 6300 + b
.
Subtract 6300 from both sides gives you:
.
-4750 = b
.
This tells us that our graph intersects the y-axis at the value -$4750. Wow. That has
to be the cost of the license, because that's how much we spent when we still have
zero hippos captured. This should agree with your graph.
.
When we substitute this value for b into the equation we get:
.
y = 900x - 4750
.
as the equation for our line.
.
Notice that where the graph crosses the x-axis the dollar value is zero. This means that
you have reached the break-even point and the value on the x-axis is the number of hippos
that you have to capture to break even. You can solve this exactly by letting y = 0 in
the equation and solving for x ... as follows:
.
0 = 900x - 4750
.
Adding 4750 to both sides of the equation results in:
.
900x = 4750
.
and dividing both sides by 900 tells you that the graph crosses the x-axis at:
.
x = 4750/900 = 5.27 hippos.
.
Hmmm! That means that when you have captured 5 hippos you still are a little behind.
But when you have captured 6 hippos you have made a little money. And when you have
captured 7 hippos you have made $1550. The problem says that you can only capture
10 hippos. And we now know that for each additional hippo we make $900 more. So what's
the maximum we can earn due to the limit of ten hippos? We can count it out. If we have
made $1550 for 7 hippos and then we capture the 8th hippo, we add $900 to the $1550 and
then have $2450. The 9th hippo gets us another $900 so we're at $3350. And finally,
the 10th hippo gets us our last $900 for a total of $4250.
.
We also could have used our equation, substituting 10 for x:
.
y = (900)*10 - 4750
.
y = 9000 - 4750 = $4250
.
Hope this helps you decipher what's going on in this problem. An the above probably
explains more about hippo hunting than you ever wanted to know. Follow it through
carefully and it may help provide some insight into linear graphs and equations.
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