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Find the dimensions of the rectangle of largest area that has its base on the x-axis and its other two vertices above the x-axis and lying on the parabola. (Round your answers to the nearest hundredth.) y = 6 - x²

Answer:

height = 4 and base = 2√2

Step-by-step explanation:

Area of a rectangle is given by;

A = base(b) × height(h)

To start off, we will first have to consider half of the rectangle.

Due to the fact that it is bounded by a parabola which is symmetric over the y-axis, it means that the rectangle will have the same area on the right and left hand sides.

So if we maximize the area of the first quadrant, it means that we are also maximizing the area of the entire rectangle.

Now, since we are dealing with the base and other 2 vertices on the x-axis, it means that the base(b) of this half rectangle will be x.

The height of this rectangle is the y coordinate of the corner that is directly above the x coordinate, which is also on the rectangle.

Therefore, the coordinates of the upper corner will be (x, y), or (x, 6 - x²). Thus our half base is x and full height is 6 - x². Area will be;

A = x(6 - x²)

A = 6x - x³

We will maximize this area by finding the derivative and equating to zero.

Thus;

A' = 6 - 3x²

At A' = 0,we have;

6 - 3x² = 0

3x² = 6

x² = 6/3

x² = 2

x = √2

Thus, from y = 6 - x², we have;

y = 6 - (√2)²

y = 6 - 2

y = 4

Since x is half base, it means our full width is 2x = 2 × √2 = 2√2

So, height = 4 and base = 2√2

8 0

3 years ago

Which shows a difference of squares? 10 y squared minus 4 x squared 16 y squared minus x squared 8 x squared minus 40 x + 25 64

evablogger [386]

Answer:

16y² - x² shows a correct difference of squares.

Step-by-step explanation:

Following are the given options from which we have to chose which represents the difference of squares.

A) 10y² - 4x²

B) 16y² - x²

C) 8x² - 40x+25

D) 64x² - 48x+9

Difference of square is given by:

a² - b² = (a + b) (a - b)

If we analyze the expressions from the choices given in the question. We should figure out that option B) should be the right answer that shows a difference of squares. i.e. 16y² - x² shows a correct difference of squares. All other options do not represent the difference of perfect squares.

As in option A) 10y² - 4x²

10y² is not a complete square of any number. Hence, option A) is incorrect.

As in option C) 8x² - 40x+25

Neither of the terms in the expression as shown in option C can be a perfect square. Hence, option C) is incorrect.

As in option D) 64x² - 48x+9

The term 48x+9 in the expression as shown in option D can not be a perfect square. Hence, option D) is incorrect.

Lets take the expression 16y² - x² of option B) and solve it.

First simplify the terms separately to associate them as perfect squares as shown below:

⇒ 16y² = (4y)² Equation (1)

⇒ x² = (1x)² Equation (2)

Now, take the difference of Equation (1) and Equation (2).

⇒ 16y² - x²

⇒ (4y)² - (1x)² Equation (3)

So,

Equation (3 )shows a difference of squares. It can also be represented as factorizing the difference of two perfect squares such as (4y - x) (4y + x).

Which is

(4y - x) (4y + x) = 16y² - x²

= (4y)² - (1x)²

So, option B) i.e. 16y² - x² shows a correct difference of squares.

Keywords: difference of square

Learn more about completing the square from brainly.com/question/12829957

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3 0

3 years ago

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Purchased additional office equipment by cheque from Jaya RM 6000 .which is debit and credit​

Purchased additional office equipment by cheque from Jaya RM 6000 .which is debit and credit