Suppose r left parenthesis d right parenthesis is a function...

Suppose R left parenthesis D right parenthesis is a function that computes the revenue generated when D car washes are sold. It is known that R left parenthesis 54 right parenthesis equals 750 and open fraction numerator d R over denominator d D end fraction close vertical bar subscript D equals 54 end subscript equals 8.20. Which of the following best interprets these results? This makes no sense, since the derivative of revenue with respect to demand must always be negative in this context. This makes no sense, since the derivative of revenue with respect to demand must be a whole number (not a decimal or fraction). When 54 car washes are sold, $750 in revenue will be made. At this level of sales, revenue increases by $8.20 per car wash. When car washes are priced at $54 each, 750 car washes will be sold. At this price point, revenue increases by 8.2 (or 8, when rounded appropriately) car washes if price stays the same.

Transcript

0:00 Question 9.

00:01 So part a in this question we are given a demand function which is relating the price and dollars of the number of units sold.

00:11 So the price rather the equation is given by b is equal to 5 minus x over 4.

00:20 Part a asks us that what is the graph of this function so let's first find out the intercepts so we have for x intercepts we have p as zero.

00:33 This means that 5 minus x over 4 is 0.

00:37 This means that x over 4 is equal to 5.

00:42 This means that x is 20.

00:44 So this is the x intercept.

00:45 And if we are talking about the y intercept, so for y intercept, x is 0, this means that p is 5.

00:52 It's keeping these things in mind if we draw a coordinate plane, obviously it will lie just in the first region because the number of units are always positive and the price must also be positive.

01:06 So we have a y intercept of 0 5 which is here.

01:10 We have an x intercept of 20, 0, which is let's say here.

01:15 So this is a straight line joining these two points.

01:19 So this is how the graph looks like.

01:21 It's a straight line.

01:23 Part b asks us that how many units can be sold when the price is 3.

01:28 So we replaced p by 3 here.

01:30 So when p is equal to three this means that five or five minus x over four is equal to three this means that x over four is five minus three which is two this means that x is equal to eight so eight units can be sold and if you want to locate then eight three will locate if this is 20 then this will be 10 then this can be eight so this is where eight and three comes part c asks us that if you want to sell 12 items then what will be the unit price so we replace x by 12 in this case so we have 5 minus 12 over 4 and 5 minus 3 is 2 so the price should be set as 2 once again we have to locate this so we have to locate 12 and 2 12 and 2 will come if over here we have 10 so that 12 will come a little bit to the right so it should come some way here so this is 12 and 2 all right so if we talk about the next part party says that find the revenue function and we have to graph the revenue function as well so the revenue function is given by x times p and the value of the equation of p is already given as 5 minus x over 4 so if you open the brackets we have 5x minus x square over 4 so this is a parabola opening downwards and its vertex is minus b over a value of b is 5.

03:03 Over a is minus 1 over 4 so it's vortex is 20 this is the vortex and it is opening and let's find out the value of wire this vortex as well so we put x is 20 over here so we have the value of revenue as 5 times 20 minus 20 square over 4 so this gives us 100 minus this will be 100 again so this will be 0 in fact so what this means is a graph will this is the coordinate axis and if we'll plot the graph it has the y intercept in fact the vertex is 20 zero so this is where the 20 zero comes in and here we have one correction this is minus b over two times a so we forgot to mention two so this would be 10 and when this is 10 this will also be replaced by 10 so the y coordinate comes out as this is 10 over here this is 10 so this becomes 50 minus this will be 25 so we get the value of the y as 25 over here all right so 10 25 lies somewhere over here let's say so the graph will be like this since it is passing through origin at x equal to zero the revenue is zero obviously so this is how the curve looks like and we are just interested in the first and this point over here the vertex is 1025.

04:41 So this is the revenue function.

04:44 Then we talk about the part e, which asks us to find the revenue and x is 2, 8 and 14.

04:52 So let's replace, this is the equation of revenue.

04:55 So let's replace x by 2, 8 and 14...

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