How can i do this math qustion: find two numbers which have a difference of 7 and a product that is a minimum?

Question

ithere are a set of question that i need to do like this. the teacher did it but i did not understand.

Answer 1

How strange.
The two guys above me had 7 and 0 at first. Then I post my answer using a negative number (-6 and 1, not right, by the way) and the second guy changes his using negatives as well. What's the point of putting an answer if you are just going to retract it later. Better yet, why not just be the first to answer, incorrectly or what not, and then change it later after everyone after you helps you answer the question?

Answer 2

First of all, you know that two numbers have a different of 7.
Let x and y be the two numbers. That would mean

x - y = 7

The product of the two numbers is given by the following formula:

P = xy

But we can actually solve one variable in terms of the other; that is,
x = 7 + y

Therefore

P = (7+y)y, OR
P = 7y + y^2

Now, notice that this is expressed in terms of one variable, so now we can declare our product function, P, as P(y).

P(y) = 7y + y^2

In order to minimize the product, we have to take the derivative and make it 0.

P'(y) = 7 + 2y
0 = 7 + 2y
-7 = 2y
y = -7/2

Since we know y = -7/2, we can solve for x.
x = 7 + y = 7 + (-7/2) = 14/2 - 7/2 = 7/2

So the two numbers are x = 7/2 and y = -7/2

If you were asked to find the minimum, all you'd have to do is solve for P(-7/2) = 7(-7/2) + (-7/2)^2 = -49/2 + (49/4) = -98/4 + 49/4 = -49/4

Answer 3

Let
x + 7 = the first number
x = the second number (so that their difference is 7)
P = their product.

Now,
P = x(x + 7)

And
P = x² + 7x

To get the minimum product, you will have to find the derivative of P. Now that takes calculus, but if you don't want to know that right now, then the derivative of P (or dP/dx) is:
dP/dx = 2x + 7

You then have to equate the derivative to zero (to get the maximum or minimum).
2x + 7 = 0

Solving for x,
x = -7/2

To check whether this is a maximum or a minimum, the second derivative should be positive at that point. The second derivative (d²P/dx²) is
d²P/dx² = 2

Now, this means that the function is positive, so it is a minimum. Therefore, the numbers are
x = -7/2 and
x + 7 = -7/2 + 7 = 7/2

Therefore, the numbers are 7/2 and -7/2. (The minimum product is -49/4)

^_^

Answer 4

Basically, the smallest 2 numbers that have a difference of 7

try 7 and 0. 7-0=7; 7*0=0

Answer 5

first, label your givens x: more beneficial # y: lesser # second, write your equations x-y=5 x-2y=9 third, multiply one or both between the equations by an excellent decision so as that a variable cancels out. I accelerated the first by -a million so as that the gadget of equations now reads -x+y=-5 x-2y=9 upload the equations -y=4 so y=-4 plug your y fee into between the unique equations to discover x x-(-4)=5 x+4=5 x=a million and there you bypass!

Answer 6

8 - 1 = 7
8 * 1 = 8

The difference is the answer in a subtraction problem, not addition.

The product is the answer in multiplication.

Answer 7

I would have to go with yupchagee's answer. Good call! Clever on the 3.5 version also.

nice.

m

Answer 8

-4 & 3; 3-(-4)=7; 3*(-4)=-12

if they don't have to be integers
-3.5 & 3.5; 3.5-(-3.5)=7; 3.5*(-3.5)=-12.25