I have a math question for you all...good luck with this one.?

Question

The demand for gasoline can be modeled as a linear function of price. If the price of gasoline is p=$1.10 per gallon, the quantity demanded is q=65 gallons. If the price rises to $1.50 per gallon, the quantity demanded falls to 45 gallons in that period. Find a formula for q in terms of p.

Answer 1

Answer: Q = -50P + 120
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Here is how I got the above answer:


*First, lets use the data to create two point sets.

(P1,Q1): (1.10, 65)
(P2,Q2): (1.50, 45)

*Now that we have two points, we have all the information we need to create a linear equation. So lets put the equation in slope-intercept form (y=mx+b: substitute Q for y and P for x).

*Start by finding slope:

m = slope = (Q2-Q1) / (P2-P1) = (45-65) / (1.50-1.10)
= -20 / 0.4 = -50

*Now use the slope to write the equation in slope-intercept form:

Q= m*P + b
Q = -50*P + b

*Now we must solve for "b", which is our Q-intercept (the "y-intercept"). To do this, plug in the values of the first point (P1,Q1) : (1.10, 65)

Q = -50*P + b
65 = -50(1.10) + b
65 = -55 + b
55 + 65 = b
120 = b

*Now that we have our Q-intercept ("y-intercept"), we plug it back into the equation:

Q = -50P + 120

Answer 2

(Jen: you missed a sign in finding the intercept. That's a negative 55, not a 55. The rest of the work is right. :-)
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It's linear, which means that the ratio of the amounts of changes will be constant. For example, if adding 10 to the first adds 20 to the second, that's always true, and adding 5 to the first will always add 10 to the second, etc.

So we know that if the price rises 40 cents, the quantity falls by 20 gallons. The number of gallons falling is half the number of gallons rising.

So if the price rises by a dollar, the quantity falls by 50 gallons.

So the slope = rise over run = change in dependent / change in independent. Here the price is independent, and quantity depends on that.

So the slope is negative fifty gallons (dependent) per dollar (independent), or -50.

If we want a full equation, we want it in slope-intercept form. The intercept we want is the y-intercept, when x (the independent variable) is zero. (Of course this is hypothetical, but we do need to find this number.)

So let's take $1.10 per gallon, and subtract a dollar from it. That increases the demand by 50 gallons. Subtract another dime from it; that increases the demand by one-tenth that, or 5 gallons. So the demand at zero dollars is 65 + 50 + 5 = 120 gallons.

Now we have a slope (-50), and an intercept (120). It's not hard to write the final equation in slope-intercept form from here: y = (slope) x + (intercept). In fact, once you do this, you can plug in the price and quantity and make sure the equation checks.

Answer 3

Since it's a linear function, I would plot it on a graph to make it easier and use gasoline as the x value and quantity demanded as the y value.

So, the two points would be (1.10, 65) and (1.50, 45).

Since now we have 2 points and we know that it's linear we can find the slope (rise over run). The "rise" was -20 (45-65) on the y axis, and the run was .40 (1.50-1.10). So, we get the slope to be (-20/.40) = -50. Then we need to find the y intercept on the graph.

We plug in one of the number sets (1.10 and 65) to the equation y=mx + b and get:

65 = (-50)*(1.10) + b
solving for b = 120.

So, we know that:

y (which we set up to be the price of gas which is p) = m*x (which we set up to be the quantity demanded which is q) + b

so the equation would be:

p = (-50)q + 120

rearranging for q would make it:

q = (p-120)/(-50)

=) I think that was what you wanted.

Answer 4

You have two points (p, q):
(1.10, 65) and (1.50, 45)
Since it is a linear function, the equation will be in the form q = m(p) + b, where m is the slope and b is the y-intercept (or p-intercept in this case).
The slope m can be found with this equation:
(q1-q2)/(p1-p2)
(65-45)/(1.10-1.50)
20/-0.4
-50
So demand drops by 50 gallons when the price of gas increases by $1.
Back to our original formula:
q = m(p) + b
q = -50(p) + b
Plug in any set of values for p and q. Let's use our first point (1.10, 65):
65 = -50(1.10) + b
65 = 55 + b
65 - 55 = b
b = 10
So our finished formula is:
q = -50(p) + 10

Answer 5

Since it is linear

q = mp + c where m is the gradient and c is a constant.

65 = m ( 1.10 ) + c
45 = m ( 1.50 ) + c

m = 65 - 45 / ( 1.1 - 1.5 )
= - 50
c = 120

So q = - 50 p + 120

You could do it the other way round, where p = mq + c
and you would get p = - 0.02 q + 2.4

then 0.02q = - p+ 2.4
Multiply by 1/0.02 to get q = - 50 p + 120 ( Same expression )

Answer 6

You have the information for two points on the line. They are defined in terms of p and q: (1.1,65) and (1.5,45).

The slope of the line is (rise)/(run) or...
(65-45)/(1.5-1.1)
20/0.4
50

The basic equation is q = mp + b where m is the slope and b is the y-intercept. You know the slope is 50 and you know at least one point on the line, so substitute for p, q, and m:
65 = 50(1.5) + b
65 = 75 + b
b = -10

Put that back into the equation and you have:
q = 50p - 10

You can plug in the two known points to make sure this is correct.

Answer 7

p+.40=q-20 p=price q=quality demand

Answer 8

f(p)= q. for every p and q that are related in the sory.

Answer 9

So, POINT-SLOPE ring any bells?