00:01
Revenue is given as a function of price.
00:03
The revenue for any price p is equal to negative 3 p squared plus 6 ,000 p.
00:13
So in part a, we just set revenue equal to zero, and now we can solve for price p.
00:22
And that will tell us what price will make the revenue is zero.
00:28
So i'm noticing that i have a common factor of p in both of these terms.
00:34
So i'm going to factor a p out to get that zero is equal to p times negative 3p plus 6 ,000.
00:46
So you have this number p times this number negative 3p plus 6 ,000, them multiplied together equals zero.
00:58
So that can only happen if p is zero or negative 3p plus 6 ,000 is zero.
01:11
And if i solve negative 3p plus 6 ,000 for p, that's negative 3p is equal to negative 6 ,000.
01:18
If i subtract 6 ,000 on both sides, then divide by negative 3, p is equal to 2 ,000 is what's left.
01:29
So if you set the price equal to 2000 or the price equal to zero, then you'll get zero revenue.
01:36
So you probably want to set the price somewhere in between there.
01:39
And the next problem b, as for what range of prices, will the revenue exceed $750 ,000? so we want revenue, which is negative 3p squared plus 6 ,000 p to be greater than 75 ,750 ,000, just like that.
02:14
All right.
02:16
So we can divide every.
02:21
Everything by negative 3 to get that p squared minus 2 ,000 p is less than negative 25 negative 250 ,000, i should say.
02:43
And notice here i multiplied everything, i multiplied everything by negative 3, or i divided everything by negative 3, i should say, because i like my coefficient on my p squared to be 1.
02:53
There's a understood 1 there.
02:55
And when i did that, i had to remember to reverse the inequality sign.
03:00
So i switched it from being greater than to less than because i multiplied both sides by a negative number.
03:04
So now we just need to find the p such that p squared minus 2 ,000 p plus 250 ,000.
03:23
It's less than zero.
03:28
And there's a couple ways we can do this.
03:31
Because the numbers are very large, i think the easiest way.
03:35
Is to take your browser, go to a website called desmos .com, go to a graphing calculator.
03:43
And i'm going to enter p squared minus 2 ,000 p plus 250 ,000.
04:05
And it doesn't like that i'm using p.
04:07
So i'm going to use x instead.
04:10
That way it won't ask me for a slider.
04:11
It'll just graph the thing.
04:13
And zoom out, zoom way out.
04:18
And you see right here, you see right here.
04:24
133 .975.
04:32
Yeah, 133 .975.
04:34
Did i make mistake? hang on just a second.
04:51
Yeah, i guess that's correct.
04:53
So if you set the price to 133 .975, it exceeds zero...