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3 years ago

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The sum of the circumference of a circle and the perimeter of a square is 24. Find the dimensions of the circle and square that

produce a minimum total area. (Let x be the length of a side of the square and r be the radius of the circle.)

1 answer:

lyudmila [28]3 years ago

8 0

Answer:

The radius of the circle is $r=1.68\ units$

The length of the square is $x=3.36\ units$

Step-by-step explanation:

we know that

The circumference of a circle is equal to $C=2\pi r$

The perimeter of the square is equal to $P=4x$

so

$24=2\pi r+4x$

Simplify

$12=\pi r+2x$

$x=(12-\pi r)/2$ -----> equation A

The area of a circle is equal to $A=\pi r^{2}$

The area of a square is $A=x^{2}$

The total area is equal to

$At=\pi r^{2}+x^{2}$ -----> equation B

substitute equation A in equation B

$At=\pi r^{2}+[(12-\pi r)/2]^{2}$

This is a vertical parabola open upward

The vertex is the minimum

The x-coordinate of the vertex is the radius of the circle that produce a minimum area

The y-coordinate of the vertex is the minimum area

Solve by graphing

The vertex is the point (1.68, 20.164)

see the attached figure

therefore

The radius of the circle is

$r=1.68\ units$

Find the value of x

$x=(12-\pi r)/2$

assume

$\pi =3.14$

$x=(12-(3.14)*(1.68))/2$

$x=3.36\ units$

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A cereal company is exploring new cylinder-shaped containers. Two possible containers are shown in the diagram.x is 6 and r=4.5

KatRina [158]

Answer:

Container $\rm X$ will have less label area than container $\rm Y$ by about $9\; \rm in$ .

Step-by-step explanation:

A rectangular sheet of paper can be rolled into a cylinder. Conversely, the lateral surface of a cylinder can be unrolled into a rectangle- without changing the area of that surface.

Indeed, the width of that rectangle will be the same as the height of the cylinder. On the other hand, the length of that rectangle should be exactly equal to the circumference of the circles on the top and the bottom of the cylinder. In other words, if a cylinder has a height of $h$ and a radius of $r$ at the top and the bottom, then its lateral surface can be unrolled into a rectangle of width $h$ and length $2\,\pi\, r$ .

Apply this reasoning to both cylinder $\mathrm{X}$ and $\rm Y$ :

For cylinder $\mathrm{X}$ , $h = 6\; \rm in$ while $r = 4.5\; \rm in$ . Therefore, when the lateral side of this cylinder is unrolled:

The width of the rectangle will be $6\; \rm in$ , while
The length of the rectangle will be $2 \, \pi \times 4.5\; \rm in = 9\, \pi\; \rm in$ .

That corresponds to a lateral surface area of $54\, \pi\; \rm in^2$ .

For cylinder $\rm Y$ , $h = 10.5\; \rm in$ while $r = 3\; \rm in$ . Similarly, when the lateral side of this cylinder is unrolled:

The width of the rectangle will be $10.5\; \rm in$ , while
The length of the rectangle will be $2\pi\times 3\; \rm in = 6\,\pi \; \rm in$ .

That corresponds to a lateral surface area of $63\,\pi \; \rm in^2$ .

Therefore, the lateral surface area of cylinder $\rm X$ is smaller than that of cylinder $\rm Y$ by $9\,\pi\; \rm in^2$ .

5 0

3 years ago

Simplify the expression please

Reika [66]

$\bf -2(3+2i)+2(4-5i)\implies 
\\\\\\
-6+8-10i-4i\implies \boxed{2-14i}$

3 0

3 years ago

Given the circle with the equation (x + 1)2 + y2 = 36, determine the location of each point with respect to the graph of the cir

kati45 [8]

$\bf \textit{equation of a circle}\\\\ 
(x- h)^2+(y- k)^2= r^2
\qquad 
center~~(\stackrel{}{ h},\stackrel{}{ k})\qquad \qquad 
radius=\stackrel{}{ r}\\\\
-------------------------------\\\\
(x+1)^2+y^2=36\implies [x-(\stackrel{h}{-1})]^2+[y-\stackrel{k}{0}]^2=\stackrel{r}{6^2}~~~~
\begin{cases}
\stackrel{center}{(-1,0)}\\
\stackrel{radius}{6}
\end{cases}$

so, that's the equation of the circle, and that's its center, any point "ON" the circle, namely on its circumference, will have a distance to the center of 6 units, since that's the radius.

$\bf ~~~~~~~~~~~~\textit{distance between 2 points}
\\\\
(\stackrel{x_1}{-1}~,~\stackrel{y_1}{0})\qquad 
A(\stackrel{x_2}{-1}~,~\stackrel{y_2}{1})\qquad \qquad 
% distance value
d = \sqrt{( x_2- x_1)^2 + ( y_2- y_1)^2}
\\\\\\
\stackrel{distance}{d}=\sqrt{[-1-(-1)]^2+(1-0)^2}\implies d=\sqrt{(-1+1)^2+1^2}
\\\\\\
d=\sqrt{0+1}\implies d=1$

well, the distance from the center to A is 1, namely is "inside the circle".

$\bf ~~~~~~~~~~~~\textit{distance between 2 points}
\\\\
(\stackrel{x_1}{-1}~,~\stackrel{y_1}{0})\qquad 
B(\stackrel{x_2}{-1}~,~\stackrel{y_2}{6})\\\\\\
\stackrel{distance}{d}=\sqrt{[-1-(-1)]^2+(6-0)^2}\implies d=\sqrt{(-1+1)^2+6^2}
\\\\\\
d=\sqrt{0+36}\implies d=6$

notice, the distance to B is exactly 6, and you know what that means.

$\bf ~~~~~~~~~~~~\textit{distance between 2 points}
\\\\
(\stackrel{x_1}{-1}~,~\stackrel{y_1}{0})\qquad 
C(\stackrel{x_2}{4}~,~\stackrel{y_2}{-8})
\\\\\\
\stackrel{distance}{d}=\sqrt{[4-(-1)]^2+[-8-0]^2}\implies d=\sqrt{(4+1)^2+(-8)^2}
\\\\\\
d=\sqrt{25+64}\implies d=\sqrt{89}\implies d\approx 9.43398$

notice, C is farther than the radius 6, meaning is outside the circle, hiking about on the plane.

3 0

3 years ago

Read 2 more answers

Triangle A C D is shown. A line is drawn from point D to point B on side A C to form a right angle. Line A D is labeled s. The l

devlian [24]

Answer:

s = 17units

Step-by-step explanation:

For this problem, we are trying to find a specific unknown side length.

We're actually given some extraneous information (information that is not needed to solve the problem): <em>It isn't necessary to know that BC is 5.</em>

If the side AD with the unknown length is part of a right triangle (the triangle in red in the attached diagram), we can use the Pythagorean Theorem to solve for AD.

It isn't clear if the diagram you were provided gives ∠ABD as a right angle, if it only gives ∠CBD as a right angle, or if it gives both as a right angle. Below, we prove that it doesn't matter, because regardless, both must be right angles.

<u>Is Triangle ABD a "right triangle"?</u>

Since B is between A and C, then the two angles ∠ABD & ∠CBD form a linear pair, and by the linear pair postulate are supplementary. Since they are supplementary, their measures add to 180°. Using the fact that all right angles are 90°, substitution, the subtraction property of equality, arithmetic, the measure of ∠ABD is also 90°, and thus must be a right angle. Thus, based on the given information, both ∠ABD & ∠CBD must be right angles.

Consequently, triangle ABD is a right triangle, by definition (it is a triangle that has a right angle).

<u>Pythagorean Theorem</u>

Since triangle ABD is a right triangle, the Pythagorean Theorem can be applied.

The Pythagorean Theorem states that $a^{2} +b^{2} =c^{2}$ where "c" is the hypotenuse (the side across from the right angle) and "a" and "b" the the lengths of the two other sides (called legs) of the right triangle. (<em>Aside: Because of the commutative property of addition, it doesn't matter which of the two legs' lengths is used for a, and which is used for b. The only thing that is required is that "c" be the length of the hypotenuse</em>)

In our triangle, side AD, with unknown length "s" is the length of our hypotenuse, and sides AB and BD are the two legs. Substituting values into the Pythagorean Theorem equation, we can solve for the unknown "s":

$a^{2} +b^{2} =c^{2}$

$(8)^{2} +(15)^{2} =(s)^{2}$

$64 +225 =s^{2}$

$289 =s^{2}$

Applying the square root property...

$\pm \sqrt{289} =\sqrt{s^{2}}$

$s=17 \text{ or } s=-17$

<u>Final Solution</u>

We discard the negative solution we obtained, since s represents the length of the side of a triangle.

s = 17units

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2 years ago

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Zepler [3.9K]

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So the final statement would be:
$(3\times10^6) + (4\times10^3) + (2\times10^1)$

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3 years ago