A spring requires 12 J12 J to stretch the spring from 8 cm8 cm to 10 cm10 cm, and an additional 48 J48 J to stretch the spring from 10 cm10 cm to 14 cm14 cm. What is the natural (unstressed) length of the spring?

6cm

Step-by-step explanation:

Given

12J to stretch from 8cm to 10cm Expression 1

Additional 48J from 10cm to 14cm Expression 2

Let l represents the length of the spring

Let k represent string constant,

We can then write ( from expression 1)

12 = integral of kx dx

The lower bound being 8 - l

And the upper bound being 10 - l

Integrating kx dx

We have

½kx²

= ½k[x²]

= ½k[(10 - l)² - (8 - l)²].

So, 12 = ½k[(10 - l)² - (8 - l)²] Equation 1

From expression 2, we can write

48 = integral of kx dx

The lower bound being 10 - l

And the upper bound being 14 - l

Integrating kx dx

We have

½kx²

= ½k[x²]

= ½k[(14 - l)² - (10 - l)²].

So, 48 = ½k[(14 - l)² - (10 - l)²] Equation 2

Divide Equation 2 by Equation 1, so we get

48/12 = ½k[(14 - l)² - (10 - l)²] / ½k[(10 - l)² - (8 - l)²] ½k cancel out ½k

So, w have

4 = [(14 - l)² - (10 - l)²] / [(10 - l)² - (8 - l)²]

Recall that a² - b² = (a + b)(a - b).

So,

4 = [(14 - l - 10 + l) (14 - l + 10 - l)] / [(10 - l - 8 + l) (10 - l + 8 - l)]

4 = [(4)(24 - 2l)]/[(2)(18-2l)]

4 = (96 - 8l)/36 - 4l) Multiply both sides by 36 - 4l

4(36 - 4l) = 96 - 8l

144 - 16l = 96 - 8l Collect Like Terms

144 - 96 = 16 - 8l

48 = 8l Divide both sides by l

48/8 = l

6 = l --- Rearrange

l = 6

So, the natural length of the string is 6cm

step-by-step explanation:

a rhombus is a parallelogram enjoying his symmetry.

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.