Substance A decomposes at a rate proportional to the amount of A present. a) Write an equation relating A to the amount left of an initial amount Upper A 0 after time t. b) It is found that 18 lb of A will reduce to 9 lb in 4.2 hr. After how long will there be only 1 lb left?

The equation relating A to the amount left of an initial amount after time t is .

After 17.5 hours there be only 1 lb left.

Given that,

Substance A decomposes at a rate proportional to the amount of A present.

We have to find,

Write an equation relating A to the amount left of an initial amount Upper A 0 after time t.

It is found that 18 lb of A will reduce to 9 lb in 4.2 hr. After how long will there be only 1 lb left?

According to the question,

1. Substance A decomposes at a rate proportional to the amount of A present.

Here,   is the amount of A present before decomposition.

A is the amount left after decomposition after time t.

The equation relating A to the amount left of an initial amount after time t is .

2. The amount of substance 18 lb of A will reduce to 9 lb in 4.2 hours.

Here, the amount of A present before decomposition is 18 lb.

And A is the amount left is 9 lb after decomposition after time t is 4.2 hours.

Therefore,

The time when only 1 lb left is,

Hence, After 17.5 hours there be only 1 lb left.

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Step-by-step explanation:

a) The equation relating A to the amount left of an initial amount Upper A 0 after time t is

A(t) = A0e^(kt)

b) It is found that 18 lb of A will reduce to 9 lb in 4.2 hr. It means that

A0 = 18

A = 9

t = 4.2

Therefore, substituting into the formula, it becomes

9 = 18e^(4.2k)

9/18 = e^(4.2k)

0.5 = e^(4.2k)

Taking ln of both sides of the equation, it becomes

Ln0.5 = ln e^(4.2k)

- 0.693 = 4.2k

k = - 0.693/4.2

k = - 0.165

The equation becomes

A(t) = 18e^(- 0.165t)

For A = 1(1 lb left), then

1 = 18e^(- 0.165t)

1/18 = e^(- 0.165t)

0.056 = e^(- 0.165t)

Taking ln of both sides, it becomes

Ln0.056 = ln e^(- 0.165t)

- 2.88 = - 0.165t

t = - 2.88/-0.165

t = 17.5 hours

After 17.5 hours, 1 lb would be left

c. 4 and 1/2 pounds is the answer! : )

intensity of the vector is v= √37 ≈ 6.08 units and make angle ∡α ≈ 9.46° with east direction

step-by-step explanation:

required vector is consists of the two componentsvx= 2+4=6 units   and vy= 1 unit and vx ⊥ vywe will use pythagorian theorem to find intensity of the vector vv∧2 = vx∧2 + vy∧2 => v = √vx∧2 + vy∧2 = √6∧2 + 1∧2 = √36+1 = √37 ≈ 6.08 unitsthe angle ∡α between vector and east direction we wil find with tanαtanα = 1/6 => α = arc tanα = arc 1/6 => α ≈ 9.46°

hope this