Precalculus. Please look at the picture
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Radioactive Cesium 137 was released into the waters off the coast of Japan and continues to leak into the Pacific Ocean. Considering that the half-life of Cesium is λ =30.17 years, create an exponential model, y=aert use it to make some predictions
y= Aert
Answers
Answer:
Exponential model
[tex]y=Ae^{rt}[/tex]
where:
y = value at "t" timeA = initial valuer = rate of growth/decayt = time (in years)Part (a)Given:
y = 100 gt = 0 yearsSubstituting given values into the formula and solving for A:
[tex]\begin{aligned}y & =Ae^{rt}\\\implies 100 & = Ae^{r \times 0}\\100 & = Ae^0\\100 & = A(1)\\A & = 100\end{aligned}[/tex]
Part (b)Given:
A = 100 gy = 50 g when t = 30.17Substituting the given values into the equation and solving for r:
[tex]\begin{aligned}y& =Ae^{rt}\\\\\implies 50 & =100e^{30.17r}\\\\\dfrac{1}{2} & = e^{30.17r}\\\\ln \dfrac{1}{2} & = \ln e^{30.17r}\\\\\ln 1-\ln2 & =30.17r \ln e\\\\0-\ln 2 & =30.17r(1)\\\\-\ln 2 & =30.17r\\\\r & = \dfrac{-\ln 2}{30.17}\end{aligned}[/tex]
Therefore, the final equation is:
[tex]y=100e^{\left(-\dfrac{\ln 2}{30.17}\right)t}[/tex]
Question 1Part (a)
Q: From 100g how much remains in 80 years?
[tex]\begin{aligned}t=80 \implies y & =100e^{\left(-\dfrac{\ln 2}{30.17}\right)80}\\& = 15.91389949 \: \sf g\end{aligned}[/tex]
Part (b)
Q: How long will it take to have 10% remaining?
10% of 100 g = 10 g
[tex]\begin{aligned}y=10 \implies 10 & =100e^{\left(-\dfrac{\ln 2}{30.17}\right)t}\\\\\dfrac{1}{10} & =e^{\left(-\dfrac{\ln 2}{30.17}\right)t}\\\\\ln \dfrac{1}{10} & =\ln e^{\left(-\dfrac{\ln 2}{30.17}\right)t}\\\\\ln 1 - \ln 10 & =\left(-\dfrac{\ln 2}{30.17}\right)t\ln e\\\\0 - \ln 10 & =\left(-\dfrac{\ln 2}{30.17}\right)t(1)\\\\-\ln 10 & =\left(-\dfrac{\ln 2}{30.17}\right)t\\\\t & = \dfrac{- \ln 10}{\left(-\dfrac{\ln 2}{30.17}\right)}\\\\t & = 100.2225706\: \sf years\end{aligned}[/tex]
Question 2Part (a)
Q: How much remains after 50 years (time)?
[tex]\begin{aligned}t=50 \implies y & =100e^{\left(-\dfrac{\ln 2}{30.17}\right)50}\\& = 31.70373153 \: \sf g\end{aligned}[/tex]
Part (b)
Q: How long to reach 20 g (amount remaining)?
[tex]\begin{aligned}y=20 \implies 20 & =100e^{\left(-\dfrac{\ln 2}{30.17}\right)t}\\\\\dfrac{1}{5} & =e^{\left(-\dfrac{\ln 2}{30.17}\right)t}\\\\\ln \dfrac{1}{5} & =\ln e^{\left(-\dfrac{\ln 2}{30.17}\right)t}\\\\\ln 1 - \ln 5 & =\left(-\dfrac{\ln 2}{30.17}\right)t\ln e\\\\0 - \ln 5 & =\left(-\dfrac{\ln 2}{30.17}\right)t(1)\\\\-\ln 5 & =\left(-\dfrac{\ln 2}{30.17}\right)t\\\\t & = \dfrac{- \ln 5}{\left(-\dfrac{\ln 2}{30.17}\right)}\\\\t & = 70.05257062\: \sf years\end{aligned}[/tex]
Related Questions
2. The bases of the trapezoid (x-14) cm and (x+2) cm. its medien is 30 cm lang, how long is the shorter
Answers
Answer:
x2-2x=58
Step-by-step explanation:
(x-14)(x+2)=30
x2+2x-14x-28=30
x2+2x-14x=28+30
x2-2x=58
A survey found that 78% of high school freshmen have Internet access at home. OF the 754 freshman at one high school, about how many would be expected to have Internet access at home?
Answers
Answer:
About 588 students.
Step-by-step explanation:
78% = 0.78
Therefore, we can conclude the following:
78% of 754 = 754 * 0.78 = 588.12
AXYZ is rotated 180°.
What are the coordinates of point X'?
Answers
Reason:
When rotating a point 180 degrees around the origin, you use the rule
[tex](x,y) \to (-x,-y)[/tex]
We flip the sign of each coordinate from positive to negative or vice versa.
The point X is at the location (-4, 1). Rotating this 180 degrees will get us to X'(4, -1)
Given f(x) = x^2 – 3x – 2, what is f(5)?
Answers
Answer:
[tex]f(5)=8[/tex]
Step-by-step explanation:
Given function: [tex]f(x)=x^2-3x-2[/tex]
To find [tex]f(5)[/tex], simply input [tex]x=5[/tex] into the function:
[tex]\begin{aligned}\implies f(5) & =5^2-3(5)-2\\& = 25-15-2\\& = 10-2\\& = 8\end{aligned}[/tex]
[tex]\huge\color{white}\boxed{\colorbox{Black}{♔︎Answer♔︎}}[/tex]
To find :-The value of f(5)
Given :-f(x) = x² - 3x - 2
Solution :-f(x) = x² - 3x - 2
we will put the value of x as 5
so that to find f(5).
f(5) = (5)² - 3×5 - 2
f(5) = 25 - 15 - 2
f(5) = 8
Result :-The value of f(5) is 8.
can someone help me rn?
Answers
Answer:
y = 1
Step-by-step explanation:
Given:
equation of the parabola: [tex]y=a(x-1)^2+q[/tex]points on the parabola: (-1, -9) and (1, 1)Substitute point (1, 1) into the given equation of the parabola to find q:
[tex]\begin{aligned}\textsf{At}\:(1,1)\implies a(1-1)^2+q &=1\\a(0)^2+q &=1\\q &=1\end{aligned}[/tex]
Substitute the found value of q and point (-1, -9) into the given equation of the parabola to find a:
[tex]\begin{aligned}\textsf{At}\:(-1,-9)\implies a(-1-1)^2+1 &=-9\\a(-2)^2+1 &=-9\\4a+1 &=-9\\4a &=-10\\a &=-\dfrac{5}{2}\end{aligned}[/tex]
Therefore, the equation of the parabola is:
[tex]y=-\dfrac{5}{2}(x-1)^2+1[/tex]
The maximum or minimum point on a parabola is the vertex.
Vertex form of a parabola: [tex]y=a(x-h)^2+k[/tex] where (h, k) is the vertex.
Therefore, the vertex of the parabola is (1, 1) and so the maximum value of y = 1
Vertex at (1,q)
There is a point already given from which parabola passes through its (1,1)
So q=1Hence
vertex=(1,1)Equation:-
y=a(x-h)²+ky=a(x-1)²+1Now put (-1,-9)
-9=a(-2)²+1-9=4a+14a=-10a=-5/2a is negative hence parabola is opening downwards
So
vertex is maximum
Max value
y=1Which of the following are based on empirical probability
Answers
The empirical probability is Life insurance, Car insurance, and Mortality.
What is empirical probability?
The ratio of the number of outcomes in which a defined event occurs to the total number of trials, not in a theoretical sample space but in a real experiment, is the empirical probability, relative frequency, or experimental probability of an event.
Therefore the empirical probability will be Life insurance, Car insurance, and Mortality.
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Sandy has built a triangular pool and wants to
add a rectangular deck around it as shown in the
figure. How many square feet of wood does Sandy
need for the deck?
Answers
Step two: Find the area of the rectangle, BH=30x15=450ft^2
Step 3: Subtract the area of the triangular pool from the area of the rectangular deck, 450-80=370ft^2
Sandy will need 370 square feet of wood to build her rectangular deck around her triangular pool.
Pls ingore all of the questions below, but help me with the attached image
What is the only number that has the same number of letters as it's meaning? ...
What number doesn't have its own Roman numeral? ...
What is the only even prime number? ...
What is the smallest perfect number? ...
What is our current numerical system based on? ...
Is Pi a rational or irrational number?
Answers
Answer:
w = 23
Step-by-step explanation:
so the total degree of a triangle is 180,
when you add all 3 it should equal 180.
35 + (5w) + (w + 7) = 180
subtract 35 from both sides, 35 - 35 + (5w) + (w + 7) = 180 - 35
(5w) + (w + 7) = 145
combine like terms (add the w's), (5w + w) + 7 = 145
6w + 7 = 145
subtract 7 from both sides, 6w + 7 - 7 = 145 - 7
6w = 138
divided both sides by 6 to get w alone, 6w / 6 = 138 / 6
w = 23
Select the expression that is equivalent to the given expression: 3(x + 2) - x
A: 5x + 6
B: 2x + 6
C: 2x + 2
D: 9
Answers
Answer:
B
Step-by-step explanation:
if you simply 3(x+2)-x, you get 2x+6.
it is not 3x+6 because x already equals 1. 2x+6=2x+6
if x=3,y=-2(negative 2) and z=6,what is 7xyz
Answers
Hope this helps!
Step-by-step explanation:
x = 3
y = - 2
z = 6
Question:[tex]7xyz[/tex]
Solution,
= 7 × 3 ( - 2 ) × 6
= 21 ( - 2 ) × 6
= - 42 × 6
= - 252
hence the answer is - 252....
Pls help me this question is bad
Answers
Answer:
using law of indices
the answer is 1
If ∠1 and ∠2 are vertical angles and m∠1 = 108°, find m∠2.
Answers
Answer:
[tex]m\angle 2= 108^{\circ}[/tex]
Step-by-step explanation:
Vertical angles are equal.
Answer:
108°
Step-by-step explanation:
vertical angles are angles opposite each other when two lines cross.
if <1 and <2 are vertical angles, it means that m<1 = m<2
m<1= 108°
therefore, m<2= 108°
Latanya has a points card for a movie theater. She receives 60 rewards points just for signing up. She earns 9.5 points for each visit to the movie theater. She needs 117 points for a free movie ticket. How many visits must Latanya make to earn a free movie ticket?
Answers
Answer:
6 visits
Step-by-step explanation:
Here the involved variables should be :
⇒ y = total number of rewards points
⇒ x = number of visits to the movie theater
=============================================================
Given :
⇒ points earned on signing up = 60
⇒ points per visit = 9.5
⇒ total points required = 117
===============================================================
Making the equation :
⇒ y = mx + b
⇒ m = points earned per visit
⇒ b = points earned on signing up
===============================================================
Solving by substitution :
⇒ 117 = 9.5x + 60
⇒ 9.5x = 117 - 60
⇒ 9.5x = 57
⇒ x = 6 visits
If Latanya receives 60 rewards points just for signing up. She earns 9.5 points for each visit to the movie theater. She needs 117 points for a free movie ticket. Then 6 visits Latanya make to earn a free ticket.
What is slope intercept form of line?The slope intercept form of a line is y=mx+b, where m is slope and b is the y intercept.
Given that,
Latanya has 60 rewards points just for signing up.
For each visit to theatre she earns 9.5 points.
To get free ticket she has to have 117 points.
We need to find how many visits she has to do to get free ticket.
We use Linear equation to solve this
Let y = total number of rewards points
x = number of visits to the movie theater
m=Points for each visit
y=mx+b
117=9.5x+60
Subtract 60 on both sides
117-60=9.5x
57=9.5x
x=6
Hence she has to make 6 visits to earn a free ticket.
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2 2/3 + 1/5
Could Someone Help??
Answers
Answer:
13/15
Step-by-step explanation:
you multiply the numbers by each other like 3×1/5 = 3/15 and 5×2/3 = 10/15
then you add them to get 13/15
in baseball Anthony hit 0.63 of the pitches thrown at him what percent of the pitches did Anthony miss
Answers
Which description matches the algebraic expression 45 n?
the sum of a number and 45
the quotient of a number and 45
the product of a number and 45
the difference of a number and 45
Answers
Answer:
The product of a number and 45
Step-by-step explanation:
Answer:
the product of a number and 45
Step-by-step explanation:
i just finished the quiz
24. Which point is NOT on the circle with center (-4,3) and radius 6? a. (2,3) b. (2,0) c. (-4,-3) d. (-10,3)
Answers
Answer:
Your answer is b. (2,0) is not a point on the circle.
Step-by-step explanation:
Center of the Circle: (h,k) = (-4,3)
Radius of the Circle: r = 6
Find the points on the circle.
(h + r ,k)
(-4 + 6 , 3) = (2,3)(h - r ,k)
(-4 - 6 , 3) = (-10,3)(h , k + r)
(-4 , 3 + 6) = (-4,9)(h , k - r)
(-4 , 3 - 6) = (-4,-3)These two shapes have the same perimeter.
regular hexagon
square
Not actual size
The length of each side of the hexagon is 8 centimetres.
Calculate the area of the square.
Question
Answers
Answer: u can use this to practice and u can also find your awnser in thes pic
Step-by-step explanation:
747.7 plus a 15% increase
Answers
Answer:
[tex]859.855[/tex]
Step-by-step explanation:
You start with the current amount:
[tex]747.7[/tex]
First calculate [tex]15\%[/tex] of [tex]747.7[/tex]:
[tex]747.7 \cdot 0.15 = 112.155[/tex]
Now add that new value back into the original amount:
[tex]747.7 + 112.155 = 859.855[/tex]
As a trick to speed things up, you can calculate it in one expression:
[tex]747.7 \cdot 1.15 = 859.855[/tex]
Please help me if i get another answer wrong i will fail pls help.
Answers
The second option reading left to right.
3^7 * 1/3^4
Steps:
Write using properties of positive and negative exponents.
You add the exponents. So (7 + -4).
3^3 = 27
We see that is not an option here. Find out which option is equal to 27.
Please help i really need this i will give a brainliest to the people that answer correctly
1st options: a positive, a negative, no
2nd options: decreases, increases, varies
Answers
Answer:
There is a negitve corrilation because as time increases speed decreases
Step-by-step explanation:
helpme pleaseee urgentlyyy
Answers
The circumference of the given circle.
Solution:In order to find the circumference, first we'll have to find the raduis.
[tex]d = 2r[/tex]
[tex] \frac{8}{2} [/tex]
[tex] = 4 \: ft[/tex]
Now, we can work out the volume.
[tex]\huge\boxed{Fomula: C = 2\pi {r}}[/tex]
[tex] c= 2 \times \pi \times 4[/tex]
[tex]c = 25.13274 \: {ft}[/tex]
[tex]\large\boxed{C = 25.13 \: ft\: (2 \: d.p)}[/tex]
Hence, the circumference of the given circle is 25.13 feets.
I need this answer asap please
Answers
Answer:283 awnser is 283
Step-by-step explanation:
Alex has 64 cubes with dimensions in ft like
1/2 x 1/2 x 1/2
He uses all the cubes to fill a box shaped like a larger rectangular prism. There are no gaps.
What is the volume in cubic ft of the larger rectangular prism?
Answers
Find the height of the tree below to the nearest foot.
Answers
(USE YOUR CALCULATOR):
Use the formula pH = log (1/H*) to find the pH of the substance given its concentration
of hydrogen ions in a liter of acid rain that has a pH. Round your answer to the nearest
ten-thousandth.
10. pH = 3.6
Answers
log(1/H*) = log(1) - log(H*)
No matter what base we’re using, log(1) = 0, so this simplifies to -log(H*). We’re also given a pH value of 3.6, so we can substitute this value into the original equation along with -log(H*) to get
3.6 = -log(H*)
Multiplying by -1 on both sides:
-3.6 = log(H*)
And rewriting in exponential form:
H* = 10^(-3.6) = 0.0003 (rounded to the nearest ten thousandth)
help does anyone know the answer to this!!! ^_^
Answers
The exponential function which represent the graph shown in image is f(x)=2.(1/4)ˣ. Option C is correct.
What is an exponential function?Exponential function is the function in which the function growth or decay with the power of the independent variable. The curve of the exponential function depends on the value of its variable.
The exponential function with dependent variable y and independent variable x can be written as,
f(x)=a.bˣ
Here, x is the variable in the power of a number, b is a constant which is known as the change factor and a is the initial value of the function.
Here, in the graph, it passes from the (0,2) and (2,0) from x and y axis. it tells us that the initial value of is 2.
a=2
The graph is plotted on right side in opposite way, which tells the value of b is in the reciprocal. Thus,
b=1/4.
Put the values as,
f(x)=a.bˣ
f(x)=2.(1/4)ˣ
Hence, the exponential function which represent the graph shown in image is f(x)=2.(1/4)ˣ. Option C is correct.
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why are asymptotes important in rational function graphs
Answers
Answer:
when sketching the curves of functions.
Step-by-step explanation:
There is a wide range of graph that contain asymptotes and that includes rational functions, hyperbolic functions, tangent curves, and more. Asymptotes are important guides when sketching the curves of functions. This is why it’s important that we know the properties, general forms, and graphs of each of these asymptotes.
Ten students received their test grades back for Math 171.
80
88
60
52
96
58
66
Find the 5 number summary.
Answers
Answer:
5 number summary hulaan mo hhahah 82 56 92 78 68
What are the quotient and remainder of (3x4 - 2x + 7x-4)= (x - 3)?
a. 3x2 + 772 +21x + 70; 206
c. 3x4 + 7%? +21x2 + 70; 206
b. 3x + 77? + 21x + 70; -206
d. 37° +73+ + 21x + 703 76
Answers
The quotient of dividing (3x⁴ - 2x³ + 7x² - 4) by (x - 3) is 3x³ + 7x² + 28x + 84 and the remainder is 248
How to determine the quotient and the remainder?The division expression is given as:
(3x⁴ - 2x³ + 7x² - 4) / (x - 3)
Set the divisor to 0
x - 3 = 0
Solve for x
x = 3
Substitute x = 3 in 3x⁴ - 2x³ + 7x² - 4
3x⁴ - 2x³ + 7x² - 4 = 3(3)⁴ - 2(3)³ + 7(3)² - 4
Evaluate
3x⁴ - 2x³ + 7x² - 4 = 248
This means that the remainder is 248
Recall that:
Dividend = Divisor * Quotient + Remainder
So, we have:
3x⁴ - 2x³ + 7x² - 4 = x - 3 * Q + 248
Subtract 248 from both sides
3x⁴ - 2x³ + 7x² - 252 = x - 3 * Q
Divide both sides by x - 3
Q = (3x⁴ - 2x³ + 7x² - 252)/(x - 3)
Factorize the numerator
Q = (x - 3)(3x³ + 7x² + 28x + 84)/(x - 3)
Cancel out the common factors
Q = 3x³ + 7x² + 28x + 84
Hence, the quotient is 3x³ + 7x² + 28x + 84 and the remainder is 248
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