7. What is the smallest number of pebbles greater than 10 for which grouping them in heaps of 7 leaves
1 extra and grouping them in heaps of 5 leaves 3 extra? Show your working.
What’s the answer

43

Step-by-step explanation:

So we want the number of pebbles p, when divided into groups of 7 has a remainder of 1 and when dividing by 5 leaves 3 extra.  

So what does this mean?  well, using division we want to say p - 1 is divisible by 7 and p - 3 is divisible by 5  

So what you want to do is either count by 7s, add one to get a new number x then subtract 3 and see if it is divisible by 5, or go the other way and count by 5s, add 3 to get x then subtract 1 and see if it is divisible by 7.

To make it into a formula use (7x +1)-3 or (5x + 3)-7.  There may be a formula but I cannot recall it.  I am going to use the formula with the 7 though, because it is easier to tell if a number is divisible by 5.

(7(1) +1)-3 = 5 but less than 10 so no

(7(2) +1)-3 = 12

(7(3) +1)-3 = 19

26

33

40  Divisible by 5!  so just add the 3 back to get 43 and that is the number of pebbles that fits the description.  it is one more than 42 which is divisible by 7

answer: the answer is 30 miles per hour.

ft/s -> mphmph = ft/s 1.466667mph = 44 divided by 1.466667mph = 30.00000

- r3ktforgood ☕

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.