Hey can anyone help with these equasions?

What real value of $t$ produces the smallest value of the quadratic $t^2 -9t - 36$?

If $s$ is a real number, then what is the smallest possible value of $2s^2 - 8s + 19$?

If $t$ is a real number, what is the maximum possible value of the expression $-t^2 + 8t -4$?

The Art of Problem Solving has begun selling a cookbook called "What Would Euler Eat?" If the price of the cookbook is $n$ dollars ($n \le 72$), then it will sell $720 - 10n$ copies. What price (in dollars) will maximize the total revenue we receive for the books?

The temperature of a point (x,y) in the plane is given by the expression $x^2 + y^2 - 4x + 2y$. What is the temperature of the coldest point in the plane?

Given that xy=3/2 and both x and y are nonnegative real numbers, find the minimum value of 10x+(3y/5)

Consider all the points in the plane that solve the equation x^2+2x=16 Find the maximum value of the product xy on this graph.

(This graph is an example of an "ellipse".)

AnonymousConfusedGuy
Oct 19, 2017

#2**0 **

I am going to take a wild guess at the first problem. **I may be wrong.**

Let's say t=1. Our problem now says, \(1^2-9x1-36\) so now we solve this to get:\( 1-9-36= -44\)

This can be considered the smallest value of the quadratic.

But if the answer needs to be positive we need t=13. So in this case \(13^2-9x13-36=16\)

Hope that helps!

Mr.Owl
Oct 19, 2017

#3**+3 **

The first few are all similar.....finding mins or maxes on parabolas

In the form ax^2 + bx +c, the min [or max] will occur at the x coordinate =

-b / [2a] ..... and we will .plug this value back into the equation to find the minimum or maximum y value

t^2 -9t - 36

This is a parabola. The smallest value will be produced when t =

-(-9) / [ 2 * 1 ] = 9/2

This is the x value of the vertex....and on this parabola,it is the minimum point

If $s$ is a real number, then what is the smallest possible value of $2s^2 - 8s + 19$?

Similar to the first one.........the smallest value will occur when s = -[8] / [ -2 *2] = 8/4 = 2

And the minimum value is 2(2)^2 - 8(2) + 19 = 8 - 16 + 19 = 11

If $t$ is a real number, what is the maximum possible value of the expression $-t^2 + 8t -4$?

Max will occur at t = - 8 / [ 2 * -1] = -8 / -2 = 4

And the max value will be -(4)^2 + 8(4) - 4 = -16 + 32 - 4 = 12

The Art of Problem Solving has begun selling a cookbook called "What Would Euler Eat?" If the price of the cookbook is $n$ dollars ($n \le 72$), then it will sell $720 - 10n$ copies. What price (in dollars) will maximize the total revenue we receive for the books?

This one is a little tricky...... price * quantity = total revenue.....so our function is

n * (720 - 10n) = 720n - 10n^2 = -10n^2 + 720n

The price that maximizes the revenue will be = - [ 720] / [ 2 * -10] = $ 36

And the max revenue is -10(36)^2 + 720(36) = $12960

The temperature of a point (x,y) in the plane is given by the expression $x^2 + y^2 - 4x + 2y$. What is the temperature of the coldest point in the plane?

Firstly...........this is a 3D object not capable of being represented in a 2D plane

We would need Calculus to solve this....the process is long and tedious.....anyway.....the minimum = -5 and it occurs at (2, -1)

Given that xy=3/2 and both x and y are nonnegative real numbers, find the minimum value of 10x+(3y/5)

Rearrange the first equation as y = 3 / [ 2x]

Putting this into the second equation, we have........ 10x + 9 / [ 10x ]

We can rewrite this as 10x + (9/10)x^-1

Take the derivative and setting it to 0.....we get....... 10 - (9/10)x^-2 = 0

Multiply through by 10 and rearrange as

100x^2 = 9 divide both sides by 100

x^2 = 9/100 take the positive root = 3/10

So....the minimum is 10(3/10) + 9 / 3 = 6 at {x,y} = {.3, 5 }

I don't understand the last problem.......sorry....!!!

CPhill
Oct 19, 2017

#6**0 **

Hey, thanks for the help but none of your answers worked so I'm still not sure how to do the problem. Is there another way?

AnonymousConfusedGuy
Oct 20, 2017

#4**+4 **

The basic technique for dealing with each of these problems is that of completing the square.

The first one, \(\displaystyle t^{2}-9t-36\).

Take a half of the coefficient of the linear term, (the 9t term), -9/2, square it, 36/4, add it to the expression and subtract it as well.

That gets you

\(\displaystyle t^{2}-9t +\frac{36}{4}-36-\frac{36}{4}\), which is still equal to the original,

and which can be written as

\(\displaystyle \left(t-\frac{9}{2}\right)^{2}-45\) .

The minimum value for that will be -45 occurring when t = 9/2, since that makes the expression inside the bracket zero. For any other value of t, the squared term will be greater than zero, making the whole expression greater than -45.

For the method to work, the coefficient of the squared term has to be 1. If the coefficient is not equal to 1, it has to be removed as a factor.

For example, if we had, (not one of yours),

\(\displaystyle -2x^{2}-8x+17\),

we would proceed as follows.

\(\displaystyle -2(x^{2}+4x)+17=-2(x^{2}+4x+4-4)+17\)

\(\displaystyle =-2\{(x+2)^{2}-4\}+17\) ,

\(\displaystyle =-2(x+2)^{2}+8+17\\=25-2(x+2)^{2}\)

Looking at that, its maximum value will be 25, occurring when x = -2, (since that makes the expression inside the bracket equal to zero).

The two variable example further down, \(\displaystyle x^{2}+y^{2}-4x+2y\), can be dealt with in much the same way.

Write it as

\(\displaystyle x^{2}-4x+y^{2}+2y\) ,

and complete the square on x and y separately. It's then easy to see what the minimum value is, and the values of x and y that give that value.

Tiggsy.

Guest Oct 19, 2017

#7**0 **

The Art of Problem Solving has begun selling a cookbook called "What Would Euler Eat?" If the price of the cookbook is n dollars n <= 72, then it will sell 720 - 10n copies. What price (in dollars) will maximize the total revenue we receive for the books?

A = n(720 - 10n)

A = -10n^2 + 720n

A = -10(n^2 - 72n)

-A/10 = n^2 - 72n

Complete the square.

-A/10 + 36^2 = n^2 - 72n + 36^2 = (n - 36)^2

Subtract 36^2 from both sides.

-A/10 = (n - 36)^2 - 36^2

and multiplying by -10 gives:

A = -10(n - 36)^2 + 12960

The total revenue is maximized when the price is $36.

I see you are taking Introduction to Algebra B...

Guest Oct 21, 2017

edited by
Guest
Oct 21, 2017