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The number of TV channels that the average U.S. home receives has been soaring in recent years. The function $t(x)=0.16 x^{2}+0.46 x+21.36$ can be used to estimate this number, where $x$ is the number of years after 1985 (Source: Nielsen Media Research, National People Meter Sample Use this function In what year did the average U.S. household receive 130 channels?

The anyone No function that the X is equal to 0.16 x it's quest plus the 2.46 x plus. Don't even where do you think here, the accept, then that number of generals and excellent blend that a number of years after a magnitude five find that involved here in the U. S. Support the people get the chance. So even put PT is equal to it. Yet they did 16 x squared +146 x bless going. People want 36 practically it put the sides, we get the 0.16 access quiet plus 0.46 x minus Safety six by 640 Now thought if Cora dictation, we use Kartik formula the quality for life X minus B plus minus On the route he square minus four a c t. But by a no. Here he quickly the 116 being the cripple 0.46 and say vehicle when a 66.64 a B and C belief formula Yet X is equal to minus 146 plus minus. The 146 is quiet. My four, the one for six a and sees my 66.64 We're doing right Bank of the Five for six. After seeing if I leave, we get X equal to manage 31.964 and X is equal to 19 point lead 214 here. Neglect negative X because number fear cannot be negative. Two X is equal to 19 ones. Go on four. Now we will find the year that it 1985 plus 19. Very cool, dude. Toe four. So, in your opinion for every U. S household to see it yet, General, that the final

Here we are given function. Yes, I think we do the one fund six Xs quit plus the open +46 x Lester even 460.6. No, if you prevent that yes. Number of candles and exit the number off year after 1985. In this question, we need to find in 40 year. Did the average U. S household received 50 channels, so we'll put PFE is equal to 50 e could put 0116 Xs square last table 46 x plus 21+16 Now the object of people's affect you get people one fund six exit square Because people by 46 Yes, minus 28.64 If you could do zero not solve this communication We use contact formula Kartik for life X equal to minus B plus minus. On the route he square my passport G c. Who was like right away. Here it was. He looked like 16 b sealed by 46 and C minus 28.6. Port would do well off a B and C in police formula. He yet it is a little mice legal 146 last minus. You feel by 46 square by four Little 1.6. I can see my 28 16 score Well, anyway. Group times legal went from six. No, After 75 days we get X is equal. Do minus 14 when 8 96 X is equal to going 0.186 Here we neglect that need to value because number off yet cannot get it. No final Ansari. Now we find that yet. So we will end well. In 1985 1995 delegates will do 1997 in this year. Every U S household received 50 General. That's the final answer.

So in this problem we are given the function that represents the number of TV channels that the average us home receives. Um since 1985. And it's asking what year did the average U. S. Household received 50 channels. So I'm gonna play 50 and four T. F. X. And what I'm gonna do is subtract 50 to the other side. So that would be 21.36 -50 which is negative 28.64 mm. And to solve this I'm gonna use the quadratic formula which I'll right over here. Okay so I'm gonna plug in A. B. And C. So that would be ax equals negative 0.46 poster minus the square root of 0.46 squared -4 times .16 -28.64 Over two times played 16. And I'm gonna go ahead and get this number under the radical first. So I have negative 160.46 plus or minus the square root of 18.54 Over a .32. And I'm gonna go ahead and take the square root is me negative point for six plus or minus 4.31 over 0.32. So I'm gonna add these together and divide by .32 and subtract them and divide by 3.32. To get my two answers. Yeah And in terms of this problem the negative answer isn't going to make a new sense since we're moving forward in years. So I'm only gonna rely on the 12.03 is my answer. So to get the year, I'm gonna add that to 1985, Which would give me 1997, Which would be the year that the average number of channels that household had in the US would be 50 channels.

What you see here is basically just a rough representation of table 13.3, which is part of this question. So for letter A were asked that if the opportunity cost of financial capital is 6% how much should the firm invest in R and D? Well, that 6% which is here on the left, right. All of these are percentages at 6%. That puts us at 400 for the value of orangey. So that's letter A. For letter B. We have to assume that the social rate of return is 2% on top of the private rate of return. All right, so that means that we have to imagine that each of these numbers has an additional 2% added onto it so that it's 14 12 10 8 and six, respectively. Okay, so if the investment is still socially optimal at 6% but now it's socially not privately optimal. Now this puts us down here and again. It may look like four, but remember that you have to add 2% to everything in the left hand column so foreign to its six and now the socially optimal level of R and D is going to be 500. So while the private optimization is at 400 the socially optimal level would be 500.