If 60% of a radioactive element remains radioactive after 400 million years, then what percent remains radioactive after 500 million years? What is the half-life of this element?
The half-life of the element is approximately 542.78 million years.
To find out what percent of the radioactive element remains after 500 million years, we can use the exponential decay formula for radioactive decay:
[tex]\[ N(t) = N_0 \times e^{-kt} \][/tex]
Where:
- [tex]\( N(t) \)[/tex] is the amount of radioactive material remaining at time \( t \)
- [tex]\( N_0 \)[/tex] is the initial amount of radioactive material
- k is the decay constant
- t is the time elapsed
Given that 60% of the radioactive element remains after 400 million years, we know that [tex]\( N(t) = 0.60N_0 \)[/tex] when [tex]\( t = 400 \)[/tex] million years. We also know that [tex]\( N_0 \)[/tex] represents the initial amount of radioactive material, which will cancel out when we're finding the ratio of remaining material, so we don't need its exact value.
Substituting these values into the exponential decay formula:
[tex]\[ 0.60N_0 = N_0 \times e^{-400k} \][/tex]
We can simplify this to find the decay constant k:
[tex]\[ 0.60 = e^{-400k} \][/tex]
Taking the natural logarithm (ln) of both sides to solve for k:
[tex]\[ \ln(0.60) = -400k \][/tex]
[tex]\[ k = \frac{\ln(0.60)}{-400} \][/tex]
Using this value of k, we can find out what percent of the radioactive element remains after 500 million years:
[tex]\[ N(500) = N_0 \times e^{-500k} \][/tex]
[tex]\[ N(500) = N_0 \times e^{-500 \times \frac{\ln(0.60)}{-400}} \][/tex]
Now, to find the half-life of the element, we know that the half-life [tex](\( T_{\frac{1}{2}} \))[/tex] is the time it takes for the amount of radioactive material to decrease by half. In other words, when [tex]\( N(t) = \frac{1}{2}N_0 \),[/tex] we have:
[tex]\[ \frac{1}{2}N_0 = N_0 \times e^{-kT_{\frac{1}{2}}} \][/tex]
[tex]\[ \frac{1}{2} = e^{-kT_{\frac{1}{2}}} \][/tex]
[tex]\[ \ln\left(\frac{1}{2}\right) = -kT_{\frac{1}{2}} \][/tex]
[tex]\[ T_{\frac{1}{2}} = \frac{\ln(2)}{k} \][/tex]
Now, we can calculate both the percentage remaining after 500 million years and the half-life of the element using the calculated value of k. Let's do that.
First, let's calculate the value of \( k \):
[tex]\[ k = \frac{\ln(0.60)}{-400} \][/tex]
[tex]\[ k ≈ \frac{-0.5108}{-400} \][/tex]
[tex]\[ k = 0.001277 \][/tex]
Now, let's use this value of k to find out what percent of the radioactive element remains after 500 million years:
[tex]\[ N(500) = N_0 \times e^{-500k} \][/tex]
[tex]\[ N(500) = N_0 \times e^{-500 \times 0.001277} \][/tex]
[tex]\[ N(500) ≈ N_0 \times e^{-0.6385} \][/tex]
Now, let's find out what percent this is of the original amount [tex](\( N_0 \))[/tex]:
[tex]\[ \frac{N(500)}{N_0} = e^{-0.6385} \][/tex]
[tex]\[ \frac{N(500)}{N_0} ≈ 0.5274 \][/tex]
So, approximately 52.74% of the radioactive element remains after 500 million years.
Now, let's calculate the half-life of the element:
[tex]\[ T_{\frac{1}{2}} = \frac{\ln(2)}{k} \][/tex]
[tex]\[ T_{\frac{1}{2}} = \frac{\ln(2)}{0.001277} \][/tex]
[tex]\[ T_{\frac{1}{2}} = \frac{0.6931}{0.001277} \][/tex]
[tex]\[ T_{\frac{1}{2}} = 542.78 \text{ million years} \][/tex]
So, the half-life of the element is approximately 542.78 million years.
Determine if the statement is true or false, and justify your answer. the product of the eigenvalues (counting multiplicities) of a is equal to the constant term of the characteristic polynomial of
a.
Lindsay draws a right triangle and adds the measures of the right angle and one acute angle. Which is a possible sum of the two angles?
Answer:
[tex]90< \gamma< 180[/tex]
Step-by-step explanation:
An acute angle is an angle that measures more than 0º and less than 90º. And a right angle is an angle that measures exactly 90º. Thus:
Let:
[tex]\alpha = Right\hspace{3}angle=90^{\circ}\\\beta= Acute\hspace{3}angle\hspace{10}0^{\circ}<\beta<90^{\circ}\\\gamma= Sum\hspace{3} of\hspace{3} the\hspace{3} two\hspace{3} angles =\alpha +\beta[/tex]
So, in this sense, the sum of the two angles can´t be equal or greater than 180º, since [tex]\beta<90^{\circ}[/tex] , also it has to be at least greater than 90º since [tex]\beta>0^{\circ}[/tex]
Therefore the sum of the two angles is:
[tex]90^{\circ}< \gamma< 180^{\circ}[/tex]
In another words, the sum is greater than 90º and less than 180º
Wich symbols makes -3/4 -11/12 a true sentence
The mathematical symbol that makes the expression -3/4 -11/12 true is subtraction.
The student is likely asking which mathematical symbol (operation) would make the expression -3/4 -11/12 a true sentence. The symbols that could apply are addition (+), subtraction (-), multiplication (x), or division (). To determine which symbol makes the sentence true, one would typically compare the result of applying each operation to the numeric values given.
For example, to test addition:
-3/4 + (-11/12) = (-9/12) + (-11/12) = -20/12 = -5/3
To test subtraction:
-3/4 - (-11/12) = (-9/12) - (-11/12) = 2/12 = 1/6
Therefore, subtraction makes the original expression a true one, because subtracting a negative is the same as adding the positive equivalent, which would simplify to an expression that equals 1/6.
Penny percent. Suppose you flip a coin 100 times, with 53 tosses landing heads up. What percentage of the tosses would be heads? What percentage would be tails?
Explain how you can use the basic moves of algebra to transform the equation 5x-3y=12 into 0=12-5x+3y
Which of the following groups have terms that can be used interchangeably?
a. critical value, probability, proportion
b. percentage, probability, proportion
c. critical value, percentage, proportion
d. critical value, percentage, probability
Final answer:
The correct answer is option b. percentage, probability, and proportion, because these terms relate to ratios or fractions of a whole and can often be used interchangeably in various statistical contexts. So the correct option is b.
Explanation:
The question is asking which of the following groups contain terms that can be used interchangeably. The correct answer is b. percentage, probability, proportion. These terms can often be used interchangeably because a percentage is a way of expressing a proportion as a number out of 100, and probability is the measure of the likelihood that an event will occur, often expressed as a decimal or a proportion. None of the other options provided (a, c, and d) contain directly interchangeable terms.
Steps:
Understand that the terms percentage, probability, and proportion all relate to ratios or fractions of a whole.Recognize that critical value is a concept used in hypothesis testing and does not equate to any of the other terms mentioned.Choose the answer that includes terms that can be used in similar contexts to express a part of a whole or likelihood.In a fraternity with 34 members , 18 take mathematics, 5 take both mathematics and physics and 8 take neither mathematics nor physics. How many take physics hit not mathematics?
how do you sketch sin, cos and tan waves?
To sketch sin, cos, and tan waves, determine the amplitude and period, plot points on a graph, and connect them smoothly.
Explanation:For a sine wave, start by determining the amplitude (A), which represents the maximum height of the wave. Then calculate the period (T) using 2π divided by the coefficient of x. Use these values to plot points on a graph and connect them smoothly to create a wave.For a cosine wave, follow the same steps as for a sine wave, but start with a different phase shift (p) value in the wave function.To sketch a tangent wave, start by finding the period using π divided by the coefficient of x. Plot points on a graph using tangent values at regular intervals and connect them.How do i find the cost per pound?
A coffee company has found that the marginal cost, in dollars per pound, of the coffee it roasts is represented by the function below, where x is the number of pounds of coffee roasted. Find the total cost of roasting 110 lb of coffee, disregarding any fixed costs.
C'(x)=- 0.018x+ 4.75, for x less than or equal to 200
The total cost of roasting 110 lb of coffee is $765.
How to solve Cost Function?Based on the given function, the marginal cost of roasting x pounds of coffee is:
C'(x) = -0.018x + 4.75 for x ≤ 200.
To find the total cost of roasting 110 lb of coffee, we need to integrate the marginal cost function to get the total cost function C(x), and then evaluate C(110).
However, since we are disregarding any fixed costs, we can assume that the total cost function is simply the integral of the marginal cost function.
Integrating C'(x) = -0.018x + 4.75 with respect to x, we get:
C(x) = -0.009x² + 4.75x + C
where C is the constant of integration. Since we are disregarding any fixed costs, we can assume that C = 0.
Therefore, the total cost of roasting 110 lb of coffee is:
C(110) = -0.009(110)² + 4.75(110) = 765
The total cost of roasting 110 lb of coffee is $765.
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Final answer:
To find the total cost of roasting 110 lb of coffee, integrate the marginal cost function C'(x) = -0.018x + 4.75 over the range of x values. Substitute 110 for x in the total cost function to find the total cost.
Explanation:
To find the total cost of roasting 110 lb of coffee, we need to integrate the marginal cost function over the desired range of x values. The marginal cost function is given by C'(x) = -0.018x + 4.75. Integrating this function will give us the total cost function C(x).
Integrating C'(x) gives C(x) = -0.009x^2 + 4.75x + C, where C is the constant of integration. Since we are disregarding any fixed costs, we can ignore the constant term C.
Now we can substitute 110 for x in the total cost function to find the total cost of roasting 110 lb of coffee: C(110) = -0.009(110)^2 + 4.75(110) = -108.9 + 522.5 = $413.6.
50PTS!
Describe the vertical asymptotes and holes for the graph of y = x-5/x^2 -1
3÷3912.00 carry to the hundreth
Carrie pays a monthly fee of $8 to stream movies online. She also pays a fee every time she rents a movie. Last month Carrie rented 6 movies, and her total bill for the month was $32.
a stadium holds 50000 people. the stadium is dived into 250 different seating section. how many seats are in each section?
A carpenter has 166 doorknobs in his workshop. Of those doorknobs, 98 are round and the rest are square. If he wants to place 7 square doorknobs in each bin, about how many bins would he need?
Answer:
about 10 bins
Step-by-step explanation:
First, we need to find out how many doorknobs are square:
166 - 98 = 68
Then, we need to divide the number of square doorknobs, 68, by the number of square doorknobs in each bin, 7: 68 ÷ 7
68 is closer to 70. So,
68 ÷ 7 is about 70 ÷ 7.
So, 68 ÷ 7 is about 10.
Jodi poured herself a cold soda that had an initial temperature 36 degrees F and immediately went outside to sunbathe where the temperature was a steady 99 degree F. After 5minutes the temperature of the soda was 46 degree F .Jodi had to run back into the house to answer the phone . What is the expected temperature of the soda after an additional 13 minutes ?
The expected temperature of the soda after an additional 13 minutes is estimated to be 72°F. This estimation is based on the assumption of a linear temperature increase, calculated from the temperature change observed during the first 5 minutes outside.
Explanation:The student's question involves the expected temperature of soda after a certain time outside in a hotter environment. Since this scenario does not involve a phase change like melting ice, but rather heating up from the surrounding air, we can infer that Newton's Law of Cooling may be applicable. The Law states that the rate of heat transfer between an object and its surroundings is proportional to the difference in temperature between them. However, without a specific rate of heat transfer given, we cannot apply a formula directly. Therefore, we might assume an approximate linear relationship based on the information provided.
Here's a possible approach:
Initial temperature of soda: 36°FTemperature of soda after 5 minutes: 46°FIncrease over 5 minutes: 46°F - 36°F = 10°FAverage rate of temperature increase: 10°F / 5 minutes = 2°F per minuteExpected additional increase after 13 minutes: 13 minutes * 2°F per minute = 26°FStarting temperature for this period: 46°FExpected temperature after an additional 13 minutes: 46°F + 26°F = 72°FThis logic assumes a simple linear rate, which might not be precisely accurate but should provide an estimated answer.
jonathan goes to th store and purchases 3 pencils for 0 .28 each, and x number of erasers for 0.38each write an expression that shows how much jonathan spent
Keywords
linear equation, variables
Step 1
Define the variables
Let
x-----> number of erasers
y-----> total cost
we know that
[tex]y=3(0.28)+(0.38)x[/tex]
[tex]y=0.84+(0.38)x[/tex] ------> this is the linear equation that represent the situation
therefore
the answer is
[tex]y=0.84+(0.38)x[/tex]
write a 5 digit number that when rounded to the nearest thousand and hundred will have a result that is the same explain
Use inductive reasoning to describe the pattern. Then find the next two numbers in the pattern. –5, –10, –20, –40, . . .
Answer:
Next two number would be –5, –10, –20,–40, –80, –160.
Step-by-step explanation:
Given : –5, –10, –20, –40, . . .
To find : Use inductive reasoning to describe the pattern. Then find the next two numbers in the pattern.
Solution : We have given that
–5, –10, –20, –40, . . .
We can see from given pattern
-5 * 2 = - 10.
- 10 * 2 = - 20 .
-20 * 2 = - 40
So, each number is multiplied by 2 to get next number.
-40 * 2 = - 80 .
- 80 * 2 = - 160 .
Therefore, Next two number would be –5, –10, –20,–40, –80, –160.
A rectangular room is 4 meters longer than it is wide, and its perimeter is 28 meters. Find the dimension of the room.
Answer: The width of the room = 5 meters
The length of the room = 9 meters
Step-by-step explanation:
Let the w denotes the width of the room , then length of the room will be :-
[tex]\text{length}=4+w[/tex]
The formula of perimeter of rectangle :-
[tex]\text{Perimeter}=2(l+w)\\\\\Rightarrow\ 28=2(4+w+w)\\\\\Rightarrow\ 28=2(4+2w)\\\\\Rightarrow\ 28 =8+4w\\\\\Rightarrow\ 4w=20\\\\\Rightarrow\ w=5[/tex]
Hence, The width of the room = 5 meters
The length of the room = 5+4 = 9 meters
The width of the room is 5 meters, and the length is 9 meters.
Explanation:To find the dimensions of the rectangular room, let's assign variables. Let x be the width of the room. Since the length is 4 meters longer than the width, we can represent the length as x + 4.
Given that the perimeter is 28 meters, we can write an equation: 2(x + 4) + 2x = 28.
Simplifying the equation, we get 4x + 8 = 28. Subtracting 8 from both sides gives 4x = 20. Dividing both sides by 4, we find x = 5. The width of the room is 5 meters, and the length is 5 + 4 = 9 meters.
the difference of 9 and the quotient of a number x and 4 is 65
Christian has $6 in his wallet and wants to spend it on apples. how many apples can Christian buy
Jonny is jogging along a track. He has already jogged 1 2/3 miles. He plans to jog a total of 3 1/4 miles. How many miles does he have left to jog?
$185 with a 6% percent markup
Harvey has a fair eight-sided die that has a different number from 1 to 8 on each side. If he rolls the die twice, what is the probability that the second number rolled is greater than or equal to the first number? Express answer as a common fraction.
The probability that the second number rolled is greater than or equal to the first number when rolling a fair eight-sided die twice is 1/16.
Explanation:To find the probability that the second number rolled is greater than or equal to the first number when rolling a fair eight-sided die twice, we need to determine the favorable outcomes and the total number of possible outcomes.
There are 8 possible outcomes for the first roll and 8 possible outcomes for the second roll. However, since the die is fair and has different numbers on each side, the second roll can only be greater than or equal to the first roll for 4 of the outcomes.
Therefore, the probability is 4 favorable outcomes out of 64 total outcomes, which simplifies to a probability of B
The amount $0.3994 rounded to the nearest cent
a recipe for 1 batch of cookies uses 3/4 cup of sugar. How many cups of sugar are used 1 1/2 batches of these cookies?
The ratio will remain constant. Then the number of cups of sugar used in 1 and 1/2 batches of these cookies will be 1.125.
What are ratio and proportion?A ratio is an ordered set of integers a and b expressed as a/b, with b never equaling 0. A percentage is a mathematical expression in which two things are equal.
A recipe for 1 batch of cookies uses 3/4 cup of sugar.
Then the number of cups of sugar used in 1 and 1/2 batches of these cookies will be
We know that the ratio will remain constant. Then we have
1 and 1/2 can be written as 3/2.
Let x be the number of cups of sugar.
Then the ratio will be
[tex]\rm \dfrac{x}{3/2} = \dfrac{3/4}{1}\\\\x = \dfrac{3 \times 3}{2 \times 4}[/tex]
Then we have
x = 9/8
x = 1.125 cups of sugar
More about the ratio and the proportion link is given below.
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The measure of the angle is fourteen times greater than its supplement
a guy walks into a store and steals $100 from register, comes back and buys $70 worth of merchandise, pys for it with same $100 and gets $30 cash back. how much did store lose?