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Brut [27]
3 years ago
5

Two numbers have a difference of 24. What is the sum of their squares if it is a minimum?

Mathematics
1 answer:
seraphim [82]3 years ago
3 0
Let "a" and "b" be some number where:

a - b = 24

We want to find where a^2 + b^2 is a minimum.  Instead of just logically figuring out that the answer is where a=b=12, I'll just use derivatives.

So we can first substitute for "a" where a = b+24

So we have (b+24)^2 + b^2 = b^2 +48b +576 + b^2
And that equals 2b^2 +48b +576

Then we take the derivative and set it equal to zero:

4b +48 = 0
4(b+12) = 0
b + 12 = 0
b = -12

Thus "a" must equal 12.

So:
a = 12
b = -12

And the sum of those two numbers squared is (12)^2 + (-12)^2 = 144 + 144 = 288.

The smallest sum is 288.
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Linen has a checking account he withdrew $130 from and ATM Tuesday Wednesday he deposited $240 Friday he wrote a check for $56 w
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Solve system with substitution
SpyIntel [72]

Answer:

The cost of 1 Rose Bush= $ 8

The cost of 1 Shrub =  $12.

Step-by-step explanation:

The cost of 11 rose bushes and 4 shrubs =  $136

The cost of 2 rose bushes and 11 shrubs =  $148

Let the cost of one rose bush =  $x

and the cost of one shrub  =  $ y

Now, according to the question:

11 x + 4 y = 136

and 2 x + 11 y  = 148

From (1), we get that 11x  = 136 - 4y

or, x = \frac{136 - 4y}{11}

Substitute this value of x in equation (2), we get

2 x + 11 y  = 148  \implies 2( \frac{136 - 4y}{11})  + 11y  = 148

or, ( \frac{272 - 8y}{11})  + 11y  = 148  \implies 272 - 8y + 121y = 1628

or, 113 y = 1356

or, y = 1356/113 = 12

⇒ y = 12, So x  =  \frac{136 - 4y}{11}  = \frac{136 -12(4)}{11} = \frac{136 - 48}{11}  = 8

or, x  =8 and  y = 12 is the solution of the above system.

Hence, the cost of 1 rose bush  = $x = $8

and  The cost of 1 shrub = $ y = $12.

4 0
3 years ago
Volume and surface area
ycow [4]

Answer:

The 1st one is 540π

The 2nd one is 1696.46

Step-by-step explanation:

To find the volume of a cylinder, you do the following:

(r)^2\pi \times h

Where r represents the radius of the circle and h represents the height. Because it's multiplication, we can change it to:

r^2\times h \times \pi

So since the radius is 6, 6^2 is 36. Then we multiply by 15, to get 540. Then now since we don't need to multiply the pi, you just leave it as 540π.

For the second one, multiply 540 by pi to get the answer, then round it to the nearest hudredth to get 1696.46.

8 0
2 years ago
Solve the Linear-Quadratic System of Inequalities. Shade the solution on the graph provided.
Pachacha [2.7K]

Answer:

We have the system:

y > x^2 - 1

y < (-1/2)*x + 3

To find the solutions of this set we need to graph the solutions range of both sets, and see the intersection between these solution ranges.

How we do it?

Start with the first one.

First, we graph the equation:

y = x^2 - 1

Now because we are using the symbol ">" means that y is smaller than the thing at the right, then the graph of the equation will be with a dashed line (which means that the points on the line are not solutions) and we will shade all the region above the line

For the other inequality we do the same:

First we graph:

y = (-1/2)*x + 3

And because we have the symbol "<" we again use a dashed line, but this time we will shade all the region below the line.

Once we shaded both regions, the region where we have both shades will be the region of solutions for the system of inequalities.

You can see the graph below.

3 0
3 years ago
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