All Answers 358232 Answers

Where is the circle sector shown??

What is the radius of the circle?

\(A_{sector}=\dfrac{\pi r^2\alpha}{360°}\\ A_{sector}=\dfrac{60}{\pi}\\ r_{sector}=\sqrt{\dfrac{60\cdot 360°}{\pi \alpha}\cdot \dfrac{2\pi}{360°}}\\ \color{blue}r_{sector}=\sqrt{\dfrac{120}{\alpha }}\)

 !

What is the area of rectangle ZWXY?

Let ZW be the baseline and WX be the height of the rectangle ZWXY, and ZW be x.

Then:

\(\overline{WX}=\dfrac{x-2}{tan\ 60°}\\ \overline{AX}=\dfrac{tan\ 30°(x-2)}{tan\ 60°}=\dfrac{x-2}{3}\\ \)

\(\triangle ZAY\\ (x-\overline{AX})^2+\overline{WX}^2=4^2 \\ (x-\dfrac{x-2}{3})^2+(\dfrac{x-2}{tan\ 60°})^2=16\\ (\dfrac{2x}{3}+\dfrac{2}{3})^2+(\dfrac{x}{tan\ 60°}-\dfrac{2}{\tan\ 60°})^2=16\\ x\in \{-4,\dfrac{32}{7}\}\ \color{blue}WolframAlpha\)

\(A_{ZWXY}=x\cdot \dfrac{x-2}{tan\ 60°}=\dfrac{32}{7}\cdot \dfrac{\frac{32}{7}-2}{tan\ 60°}\\ \color{blue}A_{ZWXY}=6.7868\)

The area of rectangle ZWXY is 6.7868.

  !

A cylinder has a height of 2 feet and a radius of 2 feet. What is its volume? Use ​𝜋 ≈ 3.14 and round your answer to the nearest hundredth.     

          Volume of a cylinder is the area of the circular base, times the height.     

          V  =  π r² h     

          V  =  3.14 • (2 ft)² • 2 ft          

          V  =  (3.14) • (4) • (2) ft³     

          V  =  25.12 ft³     

_.     

why is a negative times a negative a positive?     

A negative times a negative is a positive because it represents a double reversal of direction on the number line.  Multiplying by a negative number essentially flips the sign of the number being multiplied.  When you multiply two negatives, you are reversing the direction twice, which brings you back to the positive direction, according to Khan Academy.     

_.     

how man plutos pheres will it take to fill earth     

Volume_Earth  =  1.083 x 10¹² cubic kilometers     

Volume_Pluto   =  6.97 x 10⁹ cubic kilometers     

I obtained these volumes from the internet.     

( 1.083 x 10¹² km³ ) / ( 6.97 x 10⁹ km³ )  =   0.1554 x 10³  =  1.554 x 10²     

In other words, it would take in the neighborhood of 155.4 Plutos to fill the Earth.     

_.     

The length of a rectangle is 6 yds longer than its width. If the perimeter of the rectangle is 48 yd, find its area.     

        width   =  w     

        length  =  w + 6     

        perimeter  =  2w + 2l  =  2w + 2(w + 6)     

                   48  =  4w + 12     

                  4w  =  36     

                    w  =  9     

                      l  =  w + 6  =  9 + 6     

                      l  =  15     

              Area   =  length times width     

                    A  =  9 • 15     

                    A  =  135     

_.     

Right triangle ABC has one leg of length 6 cm, one leg of length 12 cm, and a right angle at A. A square has one side on the hypotenuse of triangle ABC and a vertex on each of the two legs of triangle ABC. What is the length of one side of the square in cm? Express your answer as a common fraction.     

c²  =  a² + b²     

hypotenuse²  =  6² + 12²     

hypotenuse  =  side of square     

side²  =  6² + 12²     

side²  =  36 + 144     

side²  =  180     

side¹  =  sqrt(180)     

side²  =  6 • sqrt(5)      The square root of 5 is an irrational number, and     

                                     therefore can't be expressed as a common fraction.     

_.     

A fence encloses a square field whose sides measure 12 m. A horse is tethered to a rope that is attached to a corner post of the field so that the horse is outside the fence, If the rope holding the horse is 10 m long, find the total grazing area of the horse in square metres.   

Area_grazing  =  π r² – ( ⁹⁰/₃₆₀ ) π r²      

                   =  (³/₄) π r²      

                   =  (³/₄) • 3.14 • (10 m)²     

                   =  (³/₄) • 3.14 • 100 m²     

                   =  235.5 m²     

The horse can graze inside a circle with a radius that's the length of the rope,     

except the 90^o portion of the circle blocked to him by the corner of the fence.     

_.     

So we know z+1/z=3

if we square that, we would get (z+1/z)^2=9

which can be simplified to z^2+2+1/z^2=9

z^2+1/z+2-2=9-2

z^2+1/z=9-2=7

In a row of five squares, each square is to be colored either red, yellow, or blue, so that no two consecutive squares have the same color, and at lest three of the squares are red.  How many ways are there to color the five squares?     

In order not to have two reds next to each other,     

the three reds must occupy squares 1, 3, and 5.     

Then you put a different color in between them.     

That leaves squares 2 and 4 to fill, and two colors     

to do it with. There are only four ways.    

Blue Blue     Yellow Yellow     Blue Yellow     Yellow Blue      

So, the answer is four ways.     

_.     

The circumference of a circle is C inches. The diameter of the circle is 19 inches. Which expression best represents the value of π ?     

circumference  =  π • diameter     

in this case,                  C   =  π • 19     

                                                C     

thererfore                       π  =  –––     

                                                19     

_.     

Teresa wants to paint the ceiling of her restaurant. The ceiling is in the shape of a square. Its side lengths are feet 55. Suppose each can of paint will cover 275 square feet. How many cans will she need to paint the ceiling?     

Area of ceiling          A_ceiling   =  55 ft • 55 ft  =  3025 ft²     

Cans needed            C_needed  =  3025 ft² / 275 cans/ft²     

                                  C_needed  =  11 cans needed     

_.