A shopkeeper earns Rs. 380 in the first week, Rs. 660 in the second week and Rs. 860 in the third week. Earning is the quadratic function of number of week for which the earning is found. a. Find the quadratic function that fits the data

9514 1404 393

  e(w) = -40w^2 +400w +20

Step-by-step explanation:

A graphing calculator or spreadsheet can do the quadratic regression and help you find the formula. Here earnings (e) as a function of weeks (w) can be expressed by ...

  e(w) = -40w^2 +400w +20

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Further development

There are a couple of other ways you can go at this. One is to write equations for a, b, c using the standard form quadratic.

  aw^2 +bw +c = e

  a +b +c = 380 . . . . . . for week 1

  4a +2b +c = 660 . . . for week 2

  9a +3b +c = 860 . . . for week 3

The solution of this system of equations will give the coefficients of the quadratic. Your graphing calculator or any of several web sites can solve these equations for you.

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Consider the differences between quadratic values for w = 1, 2, 3.

  e2 -e1 = (4a +2b +c) -(a +b +c) = 3a +b

  e3 -e2 = (9a +3b +c) -(4a +2b +c) = 5a +b

Then the second differences are ...

  (e3 -e2) -(e2 -e1) = (5a +b) -(3a +b) = 2a

In the given set of numbers, the first differences are ...

  660 -380 = 280 . . . . . . . call this d1

  860 -660 = 200

And the second difference is ...

  200 -280 = -80 . . . . . . . call this d2

From the above, we know that 2a = -80  ⇒  a = -40

and 3a +b = 280 = 3(-40) +b  ⇒  b = 400

Finally, a+b+c = 380 = -40 +400 +c  ⇒  c = 20

Then the generic solution is ...

  a = d2/2

  b = d1 -3a

  c = e1 -a -b . . . . . where e1 is the first-week's earnings

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.