How many 4-digit multiples of 21 and 15 are there and 4-digit multiples of either 21 or 15 are there?

943

Step-by-step explanation:

21| 1000

   |

       47 | 13

21-13=8

1000+8=1008

21|9999

  |

     476|3

9999-3=9996

1008,1029,...,9996

let n be number of terms

a=1008,d=21,l=9996

9996=1008+(n-1)21

9996-1008=(n-1)21

8988=(n-1)21

n-1=8988/21

n-1=428

n=428+1=429

again 15|1000

            |

                66|10

15-10=5

1000+5=1005

15|9999

  |

      666|9

9999-9=9990

1005,1020,...,9990

a=1005,d=15,l=9990

let n be number of terms.

9990=1005+(n-1)15

9990-1005=(n-1)15

8985=(n-1)15

n-1=8985/15

n-1=599

n=599+1=600

21=3×7

15=3×5

L.C.M.=3×5×7=105

105)1000(9

      -945

         55

105-55=50

1000+50=1050

105)9999(

     -945

         549

       - 525

            24

9999-24=9975

1050,1155,...,9975

let n be number of terms.

d=105

l=a+(n-1)d

9975=1050+(n-1)105

9975-1050=(n-1)105

8025=(n-1)105

n-1=8925/105

n-1=85

n=85+1=86

so there are 86 multiples of 21 and 15

(A∪B)=A+B-(A∩B)

multiples of 21 or 15=429+600-86=1029-86=943

25.2

step-by-step explanation:

2+4x6-8/10= 25.2

answer: ratio of volume of cone (with 3 times the height and equal diameter) to volume of cylinder is 2 to 1. the value of π is not used.

step-by-step explanation:

radius is diameter/2 = 4.

area of circle is π r^2.

volume of cylinder is height × area of base = hπr^2 = 3(4^2) π = 48π cubic inches.

not knowing the formula for volume of cone, i calculate it using calculus (which is for calculating things).

volume of cone is integral 0 to h area(y) dy. area(y) is π r(y)^2. r(y) runs r to zero as y runs zero to h, r(0) = r, r(h) = 0.

r(y) = r - r/h y

area(y) = π (r - r/h y)^2 = π (r^2 - 2r^2y/h + r^2y^2/h^2) = π r^2(1-2y/h+y^2/h^2)

antiderivative area(y) = π r^2 (y-y^2/h+y^3/(3h^2))

integral 0 to h = π r^2(h-h^2/h+h^3/(3h^2) - (0)) = π r^2(h - h + h/3) = π r^2(h/3) = π/3 r^2 h

volume of this cone is π/3(4^2)18 cubic inches.

= 6 16 π = 96π cubic inches.

cone has twice the volume, 96/48.

cone height is 6 times the cylinder.

ratio of volume formula: π r^2 cancels.

cone over cylinder ratio is 6h/3 to h, 2 to 1.