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Hitman42 [59]
3 years ago
6

Suppose after 2500 years an initial amount of 1000 grams of a radioactive substance has decayed to 75 grams. What is the half-li

fe of the substance? The half-life is:_______.
(A) Less than 600 years
(B) Between 600 and 700 years
(C) Between 700 and 800 years
(D) Between 800 and 900 years
(E) More than 900 years
Mathematics
1 answer:
krok68 [10]3 years ago
5 0

Answer:

The correct answer is:

Between 600 and 700 years (B)

Step-by-step explanation:

At a constant decay rate, the half-life of a radioactive substance is the time taken for the substance to decay to half of its original mass. The formula for radioactive exponential decay is given by:

A(t) = A_0 e^{(kt)}\\where:\\A(t) = Amount\ left\ at\ time\ (t) = 75\ grams\\A_0 = initial\ amount = 1000\ grams\\k = decay\ constant\\t = time\ of\ decay = 2500\ years

First, let us calculate the decay constant (k)

75 = 1000 e^{(k2500)}\\dividing\ both\ sides\ by\ 1000\\0.075 = e^{(2500k)}\\taking\ natural\ logarithm\ of\ both\ sides\\In 0.075 = In (e^{2500k})\\In 0.075 = 2500k\\k = \frac{In0.075}{2500}\\ k = \frac{-2.5903}{2500} \\k = - 0.001036

Next, let us calculate the half-life as follows:

\frac{1}{2} A_0 = A_0 e^{(-0.001036t)}\\Dividing\ both\ sides\ by\ A_0\\ \frac{1}{2} = e^{-0.001036t}\\taking\ natural\ logarithm\ of\ both\ sides\\In(0.5) = In (e^{-0.001036t})\\-0.6931 = -0.001036t\\t = \frac{-0.6931}{-0.001036} \\t = 669.02 years\\\therefore t\frac{1}{2}  \approx 669\ years

Therefore the half-life is between 600 and 700 years

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Quadrilateral ABCD has vertices A(1,0) B(5,0) C (7,2) D(3,2). Use slope to prove that ABCD is a parallelogram.
Galina-37 [17]

Answer:

AD \parallel BC

AB \parallel CD

Step-by-step explanation:

The vertices of quadrilateral ABCD are A(1,0) B(5,0) C (7,2) D(3,2).

The slope of side AB is

M_{AB}=\frac{0-0}{5-1} =\frac{0}{4}=0

The slope of side BC is

M_{BC}=\frac{2-0}{7-5} =\frac{2}{2}=1

The slope of side CD is

M_{CD}=\frac{2-2}{3-7} =\frac{0}{-4}=0

The slope of AD is

M_{AD}=\frac{2-0}{3-1} =\frac{2}{2}=1

AD \parallel BC

AB \parallel CD

We see that the opposite sides of the quadrilateral ABCD are equal.

Hence the quadrilateral is a parallelogram

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3 years ago
Help please I'm trying to get the answer for 3 3/5 divided by 1/6
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Answer:

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Step-by-step explanation:

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Solve 1/2x + 3 > 2(x-5/2)
Likurg_2 [28]

Answer:

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Step-by-step explanation:

write the division as a fraction

1/2x + 3/2x (x - 5 ÷ 2)

factor out 1/2 from the expression.

1/2x(x + 3(x - 5 ÷ 2))

divide the numbers.

1/2x(x + 3(x - 2.5))

distribute 3 through the parentheses.

1/2x(x + 3x - 7.5)

collect like terms.

1/2x(4x - 7.5)

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factor out 1/2 from the expression.

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Please i have 15 minutes​
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Answer:

x = \dfrac{-\log 7}{\log 7 - \log 2}

Step-by-step explanation:

2^x = 7^{x + 1}

Take the log of both sides.

\log 2^x = \log 7^{x + 1}

Use properties of log.

x \log 2 = (x + 1) \log 7

x \log 2 = x \log 7 + \log 7

x \log 2 - x \log 7 = \log 7

x(\log 2 - \log 7) = \log 7

x = \dfrac{\log 7}{\log 2 - \log 7}

x = \dfrac{\log 7}{-(\log 7 - \log 2)}

x = \dfrac{-\log 7}{\log 7 - \log 2}

3 0
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