Answer:
[tex]\displaystyle k = \frac{1}{32}\ln\frac{1}{2} \approx -0.02166[/tex]
Step-by-step explanation:
The decay formula is given by:
[tex]\displaystyle P(t) = P_0e^{kt}[/tex]
Where k is some constant, P₀ is the initial population, and t is the number of years.
Because the substance has a half-life of 32 years, P(t) = 1/2P₀ when t = 32. Substitute:
[tex]\displaystyle \frac{1}{2}P_0 = P_0 e^{k(32)}[/tex]
Solve for k:
[tex]\displaystyle \begin{aligned} \frac{1}{2} & = e^{32k} \\ \\ \ln\left(e^{32k}\right) & = \ln\left(\frac{1}{2}\right) \\ \\ 32k & = \ln\frac{1}{2} \\ \\ k & = \frac{1}{32}\ln\frac{1}{2} \\ \\ &\approx -0.02166\end{aligned}[/tex]
In conclusion, the value of k is about -0.02166.
The value of k in the exponential decay formula is k = 0.022
How to find the value of k?To find the constant k in the decay formula for the radioactive substance, we can use the formula for exponential decay, which is given by:
N(t) = N₀*exp(-kt)
where:
N(t) is the amount of the radioactive substance at time t.N₀ is the initial amount of the substance at t = 0 (time of measurement).e is the base of the natural logarithm (approximately 2.71828).k is the decay constant we need to find.t is the time elapsed (in this case, t is measured in years since the half-life is given in years).The half-life of the substance is the time it takes for half of the substance to decay, which is 32 years in this case. So, after one half-life, N(t) = N₀ / 2.
Now, let's set up the equation for one half-life:
N(t) = N₀ * exp(-k * 32)
Since N(t) = N₀ / 2 after one half-life, we can write:
N₀ / 2 = N₀ * e^(-k * 32)
Now, divide both sides by N₀:
1/2 = exp(-k * 32)
To solve for k, take the natural logarithm (ln) of both sides:
ln(1/2) = -k * 32
k = ln(1/2) / -32
k = 0.022
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the three sides of a triangle are 11/3 cm , 15/4 cm and 26/9 cm . find it's perimeter .
[tex] \sf\longmapsto \: \frac{371}{36} [/tex]
☘︎ Solution :- Given : Three sides are 11/3 cm , 15/4 cm and 26/9 cm To find : perimeter of the trianglePerimeter of a triangle = Sum of the length of the three sides
perimeter of the given triangle = 11/3 + 15/4 + 26/9
Make the denominator same by taking the LCM of the denominators .( LCM of 3 , 4 and 9 is 36 )
[tex]\sf\longmapsto \: \frac{132 \: + \: 135 \: + \: 104}{36} \: \: \: [/tex]
[tex]\sf\longmapsto \: \frac{371}{36} \: cm \: [/tex]
♧ Therefore, the perimeter of a triangle is 371/36 cm
Calculus Problem
1. Find the volume of the solid whose base is bounded by the graphs of y = 8 - x^2and y = x^2,
with the indicated cross sections perpendicular to the x-axis:
a. Squares
b. Semi-Circles
C. Equilateral Triangles
The two parabolas intersect for
[tex]8-x^2 = x^2 \implies 2x^2 = 8 \implies x^2 = 4 \implies x=\pm2[/tex]
and so the base of each solid is the set
[tex]B = \left\{(x,y) \,:\, -2\le x\le2 \text{ and } x^2 \le y \le 8-x^2\right\}[/tex]
The side length of each cross section that coincides with B is equal to the vertical distance between the two parabolas, [tex]|x^2-(8-x^2)| = 2|x^2-4|[/tex]. But since -2 ≤ x ≤ 2, this reduces to [tex]2(x^2-4)[/tex].
a. Square cross sections will contribute a volume of
[tex]\left(2(x^2-4)\right)^2 \, \Delta x = 4(x^2-4)^2 \, \Delta x[/tex]
where ∆x is the thickness of the section. Then the volume would be
[tex]\displaystyle \int_{-2}^2 4(x^2-4)^2 \, dx = 8 \int_0^2 (x^2-4)^2 \, dx \\\\ = 8 \int_0^2 (x^4-8x^2+16) \, dx \\\\ = 8 \left(\frac{2^5}5 - \frac{8\times2^3}3 + 16\times2\right) = \boxed{\frac{2048}{15}}[/tex]
where we take advantage of symmetry in the first line.
b. For a semicircle, the side length we found earlier corresponds to diameter. Each semicircular cross section will contribute a volume of
[tex]\dfrac\pi8 \left(2(x^2-4)\right)^2 \, \Delta x = \dfrac\pi2 (x^2-4)^2 \, \Delta x[/tex]
We end up with the same integral as before except for the leading constant:
[tex]\displaystyle \int_{-2}^2 \frac\pi2 (x^2-4)^2 \, dx = \pi \int_0^2 (x^2-4)^2 \, dx[/tex]
Using the result of part (a), the volume is
[tex]\displaystyle \frac\pi8 \times 8 \int_0^2 (x^2-4)^2 \, dx = \boxed{\frac{256\pi}{15}}}[/tex]
c. An equilateral triangle with side length s has area √3/4 s², hence the volume of a given section is
[tex]\dfrac{\sqrt3}4 \left(2(x^2-4)\right)^2 \, \Delta x = \sqrt3 (x^2-4)^2 \, \Delta x[/tex]
and using the result of part (a) again, the volume is
[tex]\displaystyle \int_{-2}^2 \sqrt 3(x^2-4)^2 \, dx = \frac{\sqrt3}4 \times 8 \int_0^2 (x^2-4)^2 \, dx = \boxed{\frac{512}{5\sqrt3}}[/tex]
Pls help me
Pls show your answer pls and thank you
Answer:
Step 1
According to given details
Number of delivered pizzas at the weekly meeting=5
Number of slices in one pizza =8
(a)
Now evaluate total number of slices
Number of slices in one pizza =8
Total slices =8x5
=40 slice
(b)
Yes, we can find more than one way
= 5 ÷ 1/8
= 5x8/1
=40/1
HELP PLS 50 PTS! To the nearest 50 square feet, what is the area of the actual store?
Question 1 options:
2,350 square feet
2,450 square feet
2,500 square feet
2,400 square feet
○ 2,350 square feet
○ 2,450 square feet
○ 2,500 square feet
● 2,400 square feet answer
Step-by-step explanation:
wellcome
rip
What is 180-90-37 as a degree?
Answer:
53 degrees
Step-by-step explanation:
since 180 is a straight line and 90 is half of that, all you need to do is take 37 away from 90 and you get 53
The equivalent degree of the of the expression is 53 degrees .
Given,
180° - 90° - 37°
Now,
The equivalent value of the expression :
180° - 90° - 37°
Firstly compute the like terms,
-90° - 37°
= -127°
Now,
180° - 127°
= 53°
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If you spin the spinner 9 times, what is the best prediction possible for the number of times it will land on yellow?
Answer:
0? There's no yellow.
Step-by-step explanation:
During its grand opening sale, a local electronics store gave customers a chance to win a new flat screen TV. Customers only had to fill out a card with their name, address, and telephone number and drop it into a box. One winner would be chosen at random. This is an example of a
Using sampling concepts, it is found that this is an example of a random sample.
How are samples classified?Samples may be classified as:
Convenient: Drawn from a conveniently available pool.Random: All the options into a hat and drawn some of them.Systematic: Every kth element is taken. Cluster: Divides population into groups, called clusters, and each element in the cluster is surveyed.Stratified: Also divides the population into groups. Then, a equal proportion of each group is surveyed.In this problem, one winner is chosen at random from all the participants, hence random sampling is used.
More can be learned about sampling concepts at https://brainly.com/question/25122507
Use the quadratic formula to find the solutions to the quadratic equation below. x^2 - 6x - 5 = 0
Step-by-step explanation:
We know the quadratic formula, which is :
[tex] \\ {\longrightarrow \qquad{ \sf{x = \frac{ - b \pm \sqrt{ {b}^{2} - 4ac } }{2a} }}} \\ \\ [/tex]
Here,
a = 1b = -6c = -5So,
[tex] \\ {\longrightarrow \qquad{ \sf{x = \frac{ - ( - 6) \pm \sqrt{ {( - 6)}^{2} - 4(1)( - 5) } }{2(1)} }}} \\ \\ [/tex]
[tex] {\longrightarrow \qquad{ \sf{x = \frac{ 6 \pm \sqrt{ 36 - 4( - 5) } }{2} }}} \\ \\ [/tex]
[tex]{\longrightarrow \qquad{ \sf{x = \frac{ 6 \pm \sqrt{ 36 + 20 } }{2} }}} \\ \\ [/tex]
[tex]{\longrightarrow \qquad{ \sf{x = \frac{ 6 \pm \sqrt{ 56 } }{2} }}} \\ \\ [/tex]
[tex]{\longrightarrow \qquad{ \sf{x = \frac{ 6 \pm \sqrt{ 2(28) } }{2} }}} \\ \\ [/tex]
[tex]{\longrightarrow \qquad{ \sf{x = \frac{ 6 \pm \sqrt{ 2(2)(14) } }{2} }}} \\ \\ [/tex]
[tex]{\longrightarrow \qquad{ \sf{x = \frac{ 6 \pm \sqrt{ 2(2)(2)(7) } }{2} }}} \\ \\ [/tex]
[tex]{\longrightarrow \qquad{ \sf{x = \frac{ 6 \pm \sqrt{ 2} . \sqrt{2} . \sqrt{2 . 7} }{2} }}} \\ \\ [/tex]
[tex]{\longrightarrow \qquad{ \sf{x = \frac{ 6 \pm 2 \sqrt{14} }{2} }}} \\ \\ [/tex]
Now, Separating the solutions :
[tex] \\ {\longrightarrow \qquad{ \sf{x = \frac{ 6 + 2 \sqrt{14} }{2} }}} \\ \\ [/tex]
[tex] {\longrightarrow \qquad{ \sf{x = 3 + \sqrt{14}}}}\\ \\ [/tex]
[tex]\\ {\longrightarrow \qquad{ \sf{x = \frac{ 6 - 2 \sqrt{14} }{2} }}} \\ \\ [/tex]
[tex] {\longrightarrow \qquad{ \sf{x = 3 - \sqrt{14}}}}\\ \\ [/tex]
Find the amount of work W done by a force F = 115i +350j
in moving an object in a straight line given by the displacement vector D = -13i+15j.
The work is equal to the dot product of the force and displacement vectors:
W = (115 i + 350 j) • (-13 i + 15 j)
W = 115 × (-13) + 350 × 15
W = 3755
3. The number 84 can be expressed as the
sum of 54 + 30. Which shows how to use
the distributive property to rewrite that
sum as a multiple of a sum whose addends
have no common factors greater than 1?
A. 2(27 + 15)
B. 3(18 + 10)
C. 5(11 + 0)
D. 6(9 + 5)
Answer:
D. 6(9 + 5)
Step-by-step explanation:
Please help me as soon as posable!! In hurry!!(24 points)
In a geometric sequence, a_ 2 = 2, a_ 3 = 20, and a_4 = 200.
Which equation can be used to find the nth term of the sequence, a_n?
A) a_n =2^n-1
B) a_n =2 · 18^n-1
C) a_n =10 · 2^n-1
D) a_n =1/5 · 10^n-1
The sequence is geometric, so
[tex]a_n = r a_{n-1}[/tex]
for some constant r. From this rule, it follows that
[tex]a_3 = r a_2 \implies 20 = 2r \implies r = 10[/tex]
and we can determine the first term to be
[tex]a_2 = r a_1 \implies 2 = 10 a_1 \implies a_1 = \dfrac15[/tex]
Now, by substitution we have
[tex]a_n = r a_{n-1} = r^2 a_{n-2} = r^3 a_{n-3} = \cdots[/tex]
and so on down to (D)
[tex]a_n = r^{n-1} a_1 = 10^{n-1} \cdot \dfrac15[/tex]
(notice how the exponent on r and the subscript on a add up to n)
write an expression for the cost of t books at 13 cents each
Answer:
13n+20 is the correct answer to ypur question
Step-by-step explanation:
NEED HELP STAT if you dont know the answer for sure then dont say it, 100 POINTS
Which statements include two quantities in the real world that are additive inverses?
Select each correct answer.
A rock climber ascends 19 feet and then descends 9 feet.
A child runs 10 feet to the right and then runs 10 feet to the left.
A person removes 14 cans from a shelf and puts 15 cans back on the shelf.
A person deposits $430 in to a bank account and then withdraws $430 from the account.
B
&
D
Step-by-step explanation:
Answer:
The Answers are A and D
Step-by-step explanation:
I took the test and got em right!
Problem Situation: Gabi buys tickets to the movies.
She buys 1 adult ticket for $14 and 3 youth tickets.
She pays a total of $35.
What is the cost of each youth ticket?
Complete the equation to represent this situation.
The letter t represents the cost of a youth ticket.
Let that be y
14+3y=35Now solve
3y=35-143y=21y=21/3y=7Each youth ticket costs 7$
Answer So you can add a number with the letter t and then add and then you can 35 dollers
This equation shows the relationship between the amount of water (w), in liters, filled in Tank A and the number of minutes (m) it took to fill it.
w = 100 + 80.5m
This table shows the relationship between the amount of water, in liters, filled in Tank B and the number of minutes it took to fill the tank
Amount of water filled in Tank B:
Minutes: Amount of water: (liters)
0 154
2 384
6 844
What is the difference, in liters, between the total amount of water filled in Tank A and Tank B after 4 minutes?
Answer:
192 liters
Step-by-step explanation:
Tank A
w = 100 + 80.5m
where:
w = water in litersm = time in minutesTherefore, when m = 4:
⇒ w = 100 + 80.5(4) = 422 liters
Tank B
Given ordered pairs: (0, 154) (2, 384) (6, 844)
[tex]\sf slope\:(m)=\dfrac{change\:in\:y}{change\:in\:x}=\dfrac{384-154}{2-0}=115[/tex]
Point-slope form of linear equation: [tex]\sf y-y_1=m(x-x_1)[/tex]
(where m is the slope and (x₁, y₁) is a point on the line)
[tex]\sf \implies y-154=115(x-0)[/tex]
[tex]\sf \implies y=115x+154[/tex]
Therefore, the equation for Tank B is:
w = 115m + 154
Therefore, when m = 4:
⇒ w = 115(4) + 154 = 614 liters
Difference
614 - 422 = 192 liters
Find equation for tank B
(0,154)(2,384)Slope:-
m=384-154/2=230/2=115Equation in point slope form
w-154=115(m)w=115m+154For tank B
w=100+80.5mPut 4on both
Tank A:-
w=100+80.5(4)w=100+322w=422LTankB
w=115(4)+154w=460+154w=614LDifference:-
614-422192LPlease help me! Thank you
Answer: 4
Step-by-step explanation: how i know by guessing
Find the length of the second base of a trapezoid with one base measuring 15 feet, a height of 7.6 feet, and an area of 98.8 square feet
answer choices
9 ft
10 ft
11 ft
12 ft
Answer:
11 feet
Explanation:
area of trapezoid: 1/2 * (a + b) * hHere given:
a = 15 feetheight = 7.6 feetarea = 98.8 feet²====================
Solving steps:
⇒ 1/2 * (15 + b) * 7.6 = 98.8
⇒ (15 + b) * 7.6 = 197.6
⇒ (15 + b) = 26
⇒ b = 11 ft
Let unknown be x
1/2(x+15.5)(7.6)=98.83.8(x+15.5}=98.8x+15.5=26x=26 .03-15.5x=10.53x≈11ftThe temperature in Denver, Colorado this
morning was -7°. In the afternoon, the
temperature rose to 16º.
What does this mean?
A. The temperature is 16° below zero
B. The temperature is 16° above zero
PLEASE HELP!!
Please HELP NEED ASAP>
Tyrone conducted an experiment with a spinner. He spun the arrow on the spinner 20 times and recorded the results in the table shown. Which statement below is a true statement based on the statement above. Please no links.
Answer:
A)
Step-by-step explanation:
The experimental probability on 'A' is higher than the theoretical probability of landing on 'A'.
All change 5. Three friends collected aluminum cans from the neighborhood to recycle. In total they collected 300 cans. Manny collected 90, Jim collected 100, and Bib collected 110. What percentage did each collect?
A. Manny 30% Jim 25% Bob 45%
B. Manny 30% Jim 33.3% Bob 36.7%
C. Manny 20% Jim 33.3% Bob 46.7%
D. Manny 18% Jim 42% Bob 50%
Please help I'll mark brainliest
Answer:
i think it b i dont know for sure
Step-by-step explanation:
arrange in ascending order. : 5/8, 3/4, 7/12, 1/2
Answer:
1/2, 7/12, 5/8, 3/4.
Step-by-step explanation:
I did that in ascending order.
Answer correctly for brainliest I dont want to rush you but i need this in like 1-2 minutes
Answer:
the correct answers are A, C, and D
Step-by-step explanation:
Those are the only ones that make sense enough!
What is the average height of 6th grade students?
And
What was the low temperature...?
Those responses will be given as numerical values
E.G. 162cm or 2 degrees Celsius (2*C)
(Edit: the first & third options are the correct ones)
Hope this helps!
Help help help help math math
Answer:
[tex]x = \geqslant 34[/tex]
Step-by-step explanation:
x is greater than or equal to 34, because the illustration shows, that it can go beyond 34. The filled in dot also means that x can also be equal to 34. (i hope i explained it in a good way)
PLEASE HELP
What can you say about the y-values of the two functions f(x) = 3x2 -3 and
g(x) = 2* -3?
A. f(x) and g(x) have equivalent minimum y-values.
B. f(x) has the smallest possible y-value.
C. The minimum y-value of g(x) approaches -3.
D. g(x) has the smallest possible y-value.
Answer:
g(x) has the smallest possible y-value of -3
Step-by-step explanation:
f(x) = 3ˣ - 3 This is an exponential graph shifted down three units. So, it has an asymptote at y = -3, which means it approaches -3 but does not touch it.Range: y > 3 (-3, ∞) g(x) = 7x² - 3 ⇒ g(x) = 7(x - 0)² - 3 This is a parabola with vertex at (0, -3) Range: y ≥ 3 [-3, ∞)
please i reallly need help right now 25 points
Answer:
D) all real numbers except x = 6
Step-by-step explanation:
If there is a solid line or dot at an x value, then it is included in the domain. Since there are open dots at x = 6, that means it is not included in the domain. With that being said, the domain can be written as:
(-∞, 6)∪(6, ∞)
Hope this helps!
Need this asap first gets brainliest
Answer:
Step-by-step explanation:
5a)→Solution,
[tex]\hookrightarrow 46+90=x[Sum\: of\: interior \:angle\: is\: equal\: to \:exterior\:angles]\\\\\hookrightarrow x=136[/tex]
5b)→Solution,
77+x=180
x=180-77
x=103
round to nearest tenth. Please solve
0.48 divide 9.23
8 divide 6.43
0.4 divide 28.08
3.5 divide 2.29
.072 divide 345
2.1 divide 1.488
thank you
Answer:
.1, 1.2, 0, 1.5, 0, 1.4 // 19.2, .8, 70.2, .7, 4791.7, .7
Step-by-step explanation:
The first set is if you meant the first number divided BY the second and the second set of numbers is if you meant it vice versa. If it was worded better I could understand better.
What is the measure, in degrees, of
Answer:
Hi good job
Step-by-step explanation:
Hector was thinking of a two-digit counting number, and he asked Simon to guess the number. Describe how you can find the probability that Simon will guess correctly on the first try.
Answer:
1/89
Step-by-step explanation:
Two digit numbers include all numbers from 10 to 99. 99-10 = 89. Therefore the probability that Simon guesses the number correctly on the first try is 1 out of 89 which can be written as 1/89.
please help!! very important!!
Answer:
d
Step-by-step explanation:
Hardware store:
4 bulbs = $20
1 bulb = $20 / 4 = $5
Grocery store:
6 bulbs = $24
1 bulb = $24 / 6 = $4
The difference in price = $5 - $4 = $1
The grocery store sells at the lesser price.
So, the grocery store charges $1 less per light bulb than the hardware store.