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56. CHALLENGE Let and be two distinct rational numbers. Find the

1 answer:

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The rational number that lies exactly halfway between a/b and c/d on a number line is x = (ad + bc)/2bd or 1/2(a/b + c/d).

<h3>How to calculate the mid-value?</h3>

To calculate the mid-value of two numbers x and y, find the average of these two numbers. I.e., (x + y)/2

Thus, the average of these two numbers is the mid-value between them.

<h3>Calculation:</h3>

It is given that,

a/b and c/d are the two rational numbers on a number line.

Consider the required rational number that lies exactly halfway between a/b and c/d as 'x'.

Then, the mid-value of these two rational numbers is

(a/b + c/d)/2

⇒ (ad + bc)/2bd ...(i)

To know that the obtained rational number is exactly halfway between a/b and c/d, consider the distance from a/b to x and c/d to x.

So, the distance from a/b to x is - "x - a/b" (x > a/b) and

the distance from x to c/d is - "c/d - x"

From the given condition, the above-obtained distances are equal

⇒ x - a/b = c/d - x

⇒ x + x = a/b + c/d

⇒ 2x = (ad + bc)/bd

⇒ x = (ad + bc)/2bd ...(ii)

From (i) and (ii), it is concluded that the required rational number is

x = (ad + bc)/2bd

Learn more about finding mid-value between two rational numbers here:

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Question: CHALLENGE: Let a/b and c/d be two distinct rational numbers. Find the rational number that lies exactly halfway between a/b and c/d on a number line.

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If you are offered one slice from a round pizza (in other words, a sector of a circle) and the slice must have a perimeter of 24

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Answer:

A 12 in diameter will reward you with the largest slice of pizza.

Step-by-step explanation:

Let $r$ be the radius and $\theta$ be the angle of a circle.

According with the graph, the area of the sector is given by

$A=\frac{1}{2}r^2\theta$

The arc lenght of a circle with radius $r$ and angle $\theta$ is $r \theta$

The perimeter of the pizza slice is composed of two straight pieces, each of length r inches, and an arc of the circle which you know has length s = rθ inches. Thus the perimeter has length

$2r+r\theta=24 \:in$

We need to express the area as a function of one variable, to do this we use the above equation and we solve for $\theta$

$2r+r\theta=24\\r\theta=24-2r\\\theta=\frac{24-2r}{r}$

Next, we substitute this equation into the area equation

$A=\frac{1}{2}r^2(\frac{24-2r}{r})\\A=\frac{1}{2}r(24-2r)\\A=12r-r^2$

The domain of the area is

$0$

To find the diameter of pizza that will reward you with the largest slice you need to find the derivative of the area and set it equal to zero to find the critical points.

$\frac{d}{dr} A=\frac{d}{dr}(12r-r^2)\\A'(r)=\frac{d}{dr}(12r)-\frac{d}{dr}(r^2)\\A'(r)=12-2r$