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3 years ago

Solve: 2 In 3 = In(x-4)

1 answer:

5 0

$\bf \begin{array}{llll} \textit{Logarithm of rationals} \\\\ \log_a\left( \frac{x}{y}\right)\implies \log_a(x)-\log_a(y) \end{array}~\hfill \begin{array}{llll} \textit{Logarithm Cancellation Rules} \\\\ log_a a^x = x\qquad \qquad \stackrel{\textit{we'll use this one}}{a^{log_a x}=x} \end{array} \\\\[-0.35em] \rule{34em}{0.25pt}$

$\bf 2\ln(3)=\ln(x-4)\implies \ln(3^2)=\ln(x-4)\implies \ln(9)=\ln(x-4) \\\\\\ \log_e(9)=\log_e(x-4)\implies e^{\log_e(9)}=e^{\log_e(x-4)}\implies 9=x-4\implies \boxed{13=x}$

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A quantity with an initial value of 7600 decays exponentially at a rate of 55% every 7 days. What is the value of the quantity a

attashe74 [19]

Answer:

1539

Step-by-step explanation:

We solve the above question using the Exponential decay formula

= A(t) = Ao(1 - r) ^t

Ao = Initial Amount invested = 7600

r = Decay rate = 55% = 0.54

t = time in weeks = 2

Hence:

A(t) = 7600(1 - 0.55)²

A(t) = 7600 × (0.45)²

A(t) = 1539

Therefore, the value of the quantity after 2 weeks is 1539

6 0

3 years ago

For every 10,000 gallons of pool water, there needs to be 1 gallon of liquid chlorine. Steve has a 55,000 gallon pool. How much

Rudik [331]

Answer:

5.5 Gallons

Step-by-step explanation:

6 0

3 years ago

X^1/6 in radical form

arsen [322]

$x^{\frac{1}{6}} = \sqrt[6]{x}$

8 0

3 years ago

Consider the case of a triangle with sides 5, 12, and the angle between them 90°.

Daniel [21]

Answer:

a) Applying Pithagoras Theorem

b) Trigonometric relations

c) No we can not

d) No we can not

e) See step-by-step explanation

f) See step-by-step-explanation

Step-by-step explanation:

a) The easiest method for calculating the missing side is applying Pithagoras Theorem

c² = a² + b²

c² = (5)² + (12)² ⇒ c² = 25 + 144 ⇒ c² = 169 ⇒ c = 13

b) The easiest method to find the missing angles, is to calculate the sin∠ and then look for arcsin fuction in tables.

sin ∠α = 5/13 sin ∠α = 0.3846 ⇒ arcsin (0.3846 ) α = 23⁰

Now we can either get the other angle subtracting 180 - 90 - 23 = 67⁰

Or calculating sinβ = 12 /13 sinβ = 0.9230 arcsin(0.9230) β = 67⁰

c) The law of cosines should not be applied to right triangles, in fact for instance in our particular case we have:

Question a)

c² = a² + b² - 2*a*b*cos90⁰ (law of cosines ) but cos 90⁰ = 0

then c² = a² + b² which is the expression for theorem of Pithagoras

In case you look for calculating the missing angles

c² = a² + b² - 2*5*13*cos α

169 = 25 + 144 - 2*5*13* cos α

As you can see 169 - 25 - 144 = 0

Then we can not apply law of cosine in right triangles, we shoul apply trigonometric relations and Pithagoras theorem, and as we saw you can get the expression of Pithagoras theorem from cosine law

c² = a² + b² - 2*a*b*cos∠90° cos ∠ 90° = 0

Then c² = a² + b²

3 0

3 years ago

Sally Sue had spent all day preparing for the prom. All the glitz and the glamour of the evening fell apart as she stepped out o

Nuetrik [128]

We have the function

$p(t)=550(1-e^{-0.039t})$

Therefore we want to determine when we have

$p(t_0)=550$

It means that the term

$e^{-0.039t}$

Must go to zero, then let's forget the rest of the function for a sec and focus only on this term

$e^{-0.039t}\rightarrow0$

But for which value of t? When we have a decreasing exponential, it's interesting to input values that are multiples of the exponential coefficient, if we have 0.039 in the exponential, let's define that

$\alpha=\frac{1}{0.039}$

The inverse of the number, but why do that? look what happens when we do t = α

$e^{-0.039t}\Rightarrow e^{-0.039\alpha}\Rightarrow e^{-1}=\frac{1}{e}$

And when t = 2α

$e^{-0.039t}\Rightarrow e^{-0.039\cdot2\alpha}\Rightarrow e^{-2}=\frac{1}{e^2}$

We can write it in terms of e only.

And we can find for which value of α we have a small value that satisfies

$e^{-0.039t}\approx0$

Only using powers of e

Let's write some inverse powers of e:

$\begin{gathered} \frac{1}{e}=0.368 \\ \\ \frac{1}{e^2}=0.135 \\ \\ \frac{1}{e^3}=0.05 \\ \\ \frac{1}{e^4}=0.02 \\ \\ \frac{1}{e^5}=0.006 \end{gathered}$

See that at t = 5α we have a small value already, then if we input p(5α) we can get

$\begin{gathered} p(5\alpha)=550(1-e^{-0.039\cdot5\alpha}) \\ \\ p(5\alpha)=550(1-0.006) \\ \\ p(5\alpha)=550(1-0.006) \\ \\ p(5\alpha)=550\cdot0.994 \\ \\ p(5\alpha)\approx547 \end{gathered}$

That's already very close to 550, if we want a better approximation we can use t = 8α, which will result in 549.81, which is basically 550.

Therefore, we can use t = 5α and say that 3 people are not important for our case, and say that it's basically 550, or use t = 8α and get a very close value.

In both cases, the decimal answers would be

$\begin{gathered} 5\alpha=\frac{5}{0.039}=128.2\text{ minutes (good approx)} \\ \\ 8\alpha=\frac{8}{0.039}=205.13\text{ minutes (even better approx)} \end{gathered}$

7 0

1 year ago