12 grams(g) of a chemical are added to a metal. The amount A, in grams, of the chemical remaining during a reaction with a metal plate decreases by 0.5g per second. If instead the plate were dissolved, the amount A', in grams, of chemical remaining would decrease by half of itself every 4 seconds. How many grams greater is A than A' after 12 seconds?

A is 4.5grams greater than A'.

Step-by-step explanation:

The initial amount of chemical, A₀ = 12g

In simple step by step terms,

If the chemical dissolves, the amount reduces by 0.5g/s.

In 12s, amount that has dissolved = 12 × 0.5 = 6g and the amount remaining, A = 12 - 6 = 6g.

If the metal dissolves, the amount reduces by half of its weight every 4 seconds. But 12 seconds has three 4 seconds.

In 4 seconds, 12 g of chemical reduces by half of 12 = 6g. And amount remaining, A' = 12 - 6 = 6g.

In another 4 seconds, 6g of chemical reduces by half of 6 = 3g. Amount of chemical remaining, A' = 6 - 3 = 3g

In the final 4 seconds, 3 g of chemical reduces by half of = 1.5g. Amount of chemical remaining finally, A' = 3 - 1.5 = 1.5g

difference in the amount left from the two cases = A - A' = 6 - 1.5 = 4.5g

If the rate of reaction analysis method to this question is required;

The two key pieces of information given in the question gives the rate of reaction in form of amount of reactant chemical remaining for the two cases.

If the chemical dissolves,

The amount of chemical left at any time is A.

The rate of reaction in g/s = -0.5g/s

dA/dt = -0.5

dA = -0.5dt

Integrating the left hand side from A₀ to A, and the right hand side from 0 to t

A - A₀ = -0.5t

A = A₀ - 0.5t

For the 2nd case if the metal plate is dissolving,

The amount of chemical left at any time is A'

Half-Life = 4 secs

k = (In 2)/(half life) = (In 2)/4 = 0.1733/s

dA'/dt = -0.1733A'

dA'/A' = -0.1733 dt

Integrating from A₀ to A' and from 0 to t

In (A'/A₀) = -0.1733t

A'/A₀ = e⁻⁰•¹⁷³³ᵗ

A' = A₀e⁻⁰•¹⁷³³ᵗ

After t = 12s,

A = A₀ - 0.5t = 12 - 0.5(12) = 12 - 6 = 6g

A' = A₀e⁻⁰•¹⁷³³ᵗ = 12e⁻⁰•¹⁷³³⁽¹²⁾ = 12 × 0.125 = 1.5g

The difference in mass = A - A' = 6 - 1.5 = 4.5g

Hope this helps!!!

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.

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step-by-step explanation: