Let's do the usual thing and make t the years since 1950. We'll just abbreviate a billion B.
f(1950-1950)=39.4 B
f(2000-1950) =2025.2 B
Our exponential form for f will be
[tex]f = a e^{kt}[/tex]
[tex]39.4 \textrm{ B} = a e^{ 0 k} = a[/tex]
[tex]2025.2 \textrm{ B} = a e^{50 k}[/tex]
Dividing
[tex]\dfrac{2025.2}{39.4} = e^{50 k}[/tex]
[tex]50 k = \ln \dfrac{2025.2}{39.4}[/tex]
[tex]k = \frac 1 {50} \ln \dfrac{2025.2}{39.4} \approx 0.0787932[/tex]
Our function is
[tex]f = 39.4 \textrm{ B } e^{0.0787932 t }[/tex]
Since [tex]e^{0.0787932} \approx 1.08198[/tex]
[tex]f = 39.4 \textrm{B } 1.08198^t }[/tex]
around 8.2 % annualized growth.
Final answer:
An exponential function modeling U.S. federal budget growth between 1950 and 2000 is based on the formula f(t) = a ×[tex]b^t[/tex], where 'a' is the initial value in 1950 ($39.4 billion) and 'b' is the growth factor, calculated from the data in 2000 ($2025.2 billion). Solving the equations gives us the function [tex]f(t) = 39.4 * (51.4^{(1/50)})^t.[/tex]
Explanation:
We can model the growth of the U.S. federal budget using an exponential function, given the data from 1950 and 2000. First, we identify the years as our variable 't' where t=0 corresponds to the year 1950. We'll use the formula for exponential growth, which is f(t) = a × b^t, where 'a' is the initial amount, 'b' is the growth factor, and 't' is the time in years.
From the question, we have two data points: the budget in 1950 (t=0) is $39.4 billion, which gives us our 'a' value. In 2000 (t=50), the budget is $2025.2 billion. Plugging these values into our exponential function, we get two equations:
f(0) = 39.4 = a × [tex]b^0[/tex] which simplifies to a = 39.4.
f(50) = 2025.2 = 39.4 × b^50.
To find the value of 'b', we solve the second equation:
2025.2 = 39.4 × [tex]b^{50[/tex]
Divide both sides by 39.4:
51.4 = [tex]b^{50[/tex]
Take the 50th root of both sides:
b =[tex]51.4^(1/50).[/tex]
Now that we know both 'a' and 'b', we can write the exponential function that models the federal budget from 1950 to 2000:
[tex]f(t) = 39.4 * (51.4^{(1/50)})^t.[/tex]
Keep in mind that this model does not account for complexities like inflation or changing economic conditions. It provides a simplified view of the growth of the federal budget over this particular time period.
HELP!!!!!! I believe its B but I am not sure! :(
Answer:
A
Step-by-step explanation:
When solving for x as an exponent, we need to use logarithms in order to undo the operation and rearrange the terms. We use log rules to bring down the exponent and solve. Logarithms are the inverse operations to exponents and vice versa. We have one special kind of logarithm called the natural logarithm whose base is e. We write it as ln. Since our base is e here, we will use the natural logarithm to rearrange and isolate x.
[tex]e^{4x-1} =3[/tex]
We begin by applying the natural logarithm to each side.
[tex]ln(e^{4x-1}) =ln(3)[/tex]
Log rules allow use to rearrange the exponent as multiplication in front of the log.
[tex](4x-1)ln(e) =ln(3)[/tex]
ln e as an inverse simplifies to 1.
[tex](4x-1)(1)=ln(3)[/tex]
We now apply the inverse operations for subtraction and multiplication.
[tex]4x-1+1=1+ln(3)\\4x=1+ln(3)\\\frac{4x}{4} =\frac{1+ln3}{4} \\x =\frac{1+ln3}{4}[/tex]
Option A is correct.
What is the answer to a+b=s+r; r
Answer: " r = a + b - s " .
_____________________________________________________
Step-by-step explanation:
_____________________________________________________
Given:
a + b = s + r ; Solve for: "r" ; that is; rewrite the equation in terms of: " r " .
_____________________________________________________
Subtract "s" from each side of the equation;
to isolate "r" on one side of the equation:
_____________________________________________________
→ a + b - s = s + r - s ;
→ to get:
→ a + b - s = r ;
____________________________________________________
↔ " r = a + b - s " .
____________________________________________________
→ which is the answer.
____________________________________________________
Hope this helps!
Best wishes to you!
____________________________________________________
Someone help me please 30 points
Answer:4.-37 5.-6 6.-33 11.-20 12.-16
13.19
Step-by-step explanation:
Answer:
1. -41
2. -40
3. -4
4. -37
5. -24
6. -33
8. -47
9. 14
10. 59
11. -20
12. -16
13. 19
25 POINTS! EXPLAIN FOR BRAINLIEST/THANK YOU.
By the polynomial remainder theorem, a polynomial [tex]p(x)[/tex] has a factor of [tex]x-c[/tex] if [tex]p(c)=0[/tex]. So all you need to do is check the value of [tex]f(x)[/tex] at [tex]x=1,3,-3,5,-5[/tex]. You should get
[tex]f(1)=-96[/tex] (so [tex]x-1[/tex] is NOT a factor)
[tex]f(3)=-96[/tex] (so [tex]x-3[/tex] is NOT a factor)
[tex]f(-3)=f(5)=f(-5)=0[/tex] (so the last three options are factors)
Answer:
the answer is c and yeah
Step-by-step explanation:
Which of the following points is a solution of the inequality y < -|x|?
A) (1, -2)
B) (1, -1)
C) (1, 0)
Answer: choice A only
(1,-2) is the only solution (from the list of choices)
====================================
Explanation:
Let's go through each answer choice. We'll plug the coordinates in one at a time.
-------------
Choice A has the point (1,-2) so x = 1 and y = -2 pair up together
y < -|x|
-2 < -|1|
-2 < -1
This is a true statement as -2 is to the left of -1 on the number line. So (1,-2) is one solution. Let's see if there are others.
-------------
Choice B) plug in (x,y) = (1,-1)
y < -|x|
-1 < -|1|
-1 < -1
False. A number is not smaller than itself. So we can cross B off the list.
-------------
Choice C) plug in (x,y) = (1,0)
y < -|x|
0 < -|1|
0 < -1
This is false because -1 is smaller than 0. Cross choice C off the list.
-------------
Only choice A is a solution point for this inequality. If we were to graph the inequality, we would see only point A is in the shaded region while the other points are outside the shaded region.
The dashed boundary line does not count as the shaded region. This visually confirms why point B does not work.
After evaluating each point against the inequality y < -|x|, only point A (1, -2) satisfies the condition, making it the correct answer.
Explanation:To determine which point is a solution for the inequality y < -|x|, we need to check if the y-value of each point is less than the negative absolute value of its corresponding x-value.
For point A (1, -2), we check if -2 < -|1|. Since |-1| = 1, we are checking if -2 < -1, which is true. So, point A is a solution.For point B (1, -1), we check if -1 < -|1|. Since |-1| = 1, we are verifying if -1 < -1, which is not true. Therefore, point B is not a solution.For point C (1, 0), we check if 0 < -|1|. Since |-1| = 1, we are verifying if 0 < -1, which is not true. Hence, point C is not a solution.Therefore, the correct answer is point A (1, -2), as it is the only point where the y-value is less than the negative absolute value of the x-value.
Which of the options represents the resulting equation after an equivalent expression for y is substituted into the second equation?
1 THROUGH 4 ARE THE ANSWERS
Answer:
Correct choice is D
Step-by-step explanation:
Given the system of two equations:
[tex]\left\{\begin{array}{l}4x+y=8\\6x-9y=12\end{array}\right..[/tex]
Fro mthe first equation
[tex]y=8-4x.[/tex]
Substitute into the second equation [tex]8-4x[/tex] instead of y:
[tex]6x-9(8-4x)=12.[/tex]
Note that [tex]8-4x=-4x+8,[/tex] then the second equation will take look
[tex]6x-9(-4x+8)=12.[/tex]
Justin's rice ball recipe uses 100 grams of rice to make 11 rice ball. Justin has 700grams of rice. How many rice balls can Justin make with 700 grams of rice?
Answer: I TINK 77 rice balls
Step-by-step explanation:
Answer:77
Step-by-step explanation:
Use the dot product to find |v| when v=(-8,2)
Answer:
2√17
Step-by-step explanation:
V = (-8, 2)
|v| = √((-8)^2 + 2^2)
= √(64 + 4)
= √68
= √4* √17
= 2√17
Answer:
= 2 sqrt(17)
Step-by-step explanation:
To find the magnitude of v using the dot product
|v| = sqrt(v1*v1 + v2*v2)
= sqrt( -8*-8 + 2*2)
= sqrt(64+ 4)
= sqrt(68)
= sqrt(4*17)
= sqrt(4) sqrt(17)
= 2 sqrt(17)
There are 64 pretzels in a 16-ounce bag of chocolate covered pretezel. What is the rate of pretzels per ounce?
Help me please lots of points
Answer:
In your first picture
7) -40
14) 200
Step-by-step explanation:
The symbol in the picture is asking for multiplication. Typically for these, teachers encourage the use of the multiplication button on a calculator, but otherwise you need to know the basic steps of multiplication.
Answer:
7) -40
14) 200
Step-by-step explanation:
Ali caught a fish that weighted 9 and 1/2 lb. Lisa caught a fish that weighed 5 and 1/ 3 lb. How much more did ali fish weight.
Final answer:
Ali's fish weighs 25/6 lb more than Lisa's fish.
Explanation:
To find out how much more Ali's fish weighs compared to Lisa's fish, we need to subtract the weight of Lisa's fish from the weight of Ali's fish.
Ali's fish weighs 9 and 1/2 lb, which can be rewritten as 19/2 lb. Lisa's fish weighs 5 and 1/3 lb, which can be rewritten as 16/3 lb.
Now, subtracting the weight of Lisa's fish from Ali's fish, we have:
Ali's fish weight - Lisa's fish weight = (19/2) lb - (16/3) lb = (57/6) lb - (32/6) lb = 25/6 lb.
Therefore, Ali's fish weighs 25/6 lb more than Lisa's fish.
Consider the scatter plot.
Scatter plot with line of best fit of y equals 0.75x plus five.
The line of best fit is y = 0.75x + 5.
Choose the best representation for the slope.
The slope of the line of best fit shows that each additional minute, the distance increases by 0.75 feet.
The slope of the line of best fit shows that each additional minute, the distance decreases by 5 feet.
The slope of the line of best fit shows that each additional minute, the distance decreases by 0.75 feet.
The slope of the line of best fit shows that each additional minute, the distance increases by 5 feet.
Answer:
The correct answer option is: The y-intercept of the line of best fit shows that when time started, the distance was 5 feet.
Step-by-step explanation:
We are given a scatter plot with a best fit line as shown on the given graph.
The equation of the best fit line is given by:
y = 0.75x + 5
So with the help of the equation and by looking at the given graph, we can conclude about the representation of the y intercept that the the y-intercept of the line of best fit shows that when time started, the distance was 5 feet.
Since the distance shown on the y axis is already 5 when the time started at 0 minutes.
List the first four terms of a geometric sequence with t 1 = 4 and t n = -3t n-1 .
[tex]t_1=4\\t_n=-3t_{n-1}\\\\t_2=-3\cdot4=-12\\t_3=-3\cdot(-12)=36\\t_4=-3\cdot36=-108[/tex]
What is an equation of the line that is parallel to y=4x-1 and that passes through the point (-3,5)?
A. y=4x-3
B. y=4x+5
C. y=4x+17
D. y=-4x+17
Answer:
C. y=4x+17
Step-by-step explanation:
If we want to find a line parallel to y =4x-1, the slopes will have to be the same because parallel lines have the same slope. Knowing y= mx+b we know the slope of the old line is 4, so the slope of the parallel line is 4.
If we know the slope of the line and a point, we can use the point slope form of a line
y-y1 = m(x-x1)
y-5 = 4(x--3)
y-5=4(x+3)
Distribute the 4
y-5 = 4x+12
Add 5 to each side
y-5+5 = 4x+12+5
y = 4x+17
Answer: The answer is C. y=4x+17
Step-by-step explanation:
The other person was correct and i got an A<3
An initial population of 8 wolves increases by 9% each year. The function f(x) = abx models this situation. Which value does x represent in the function when determining the population after 6 years?
A value which x represent in the function when determining the population after 6 years is 6.
In Mathematics and Geometry, an exponential function can be modeled by using this mathematical equation:
[tex]f(x)=a(b)^x[/tex]
Where:
a represents the initial value or y-intercept.x represents the time or x-variable.b represents the rate of change or common ratio.In this context, we can reasonably infer and logically deduce that the variable x in the standard form of the equation of an exponential function represent time, which can either be measure in years, months, days, hours, minutes, or seconds.
In conclusion, x represents 6 in the exponential function when determining the population after 6 years.
Complete Question:
An initial population of 8 wolves increases by 9% each year. The function [tex]f(x) = ab^x[/tex] models this situation. Which value does x represent in the function when determining the population after 6 years?
Evaluate the infinite sum
Answer:
It does not converge.
Step-by-step explanation:
Given is an infinite geometric series whose first term is a = 4/7 and common ratio is r = 7/6.
The series ∑a·rⁿ converges if we have |r| < 1.
And the series ∑a·rⁿ diverges if we have |r| > 1.
But we can easily check that |r| = 7/6 > 1.
It means the given series diverges, i.e. does not converge.
Hence, option D is correct answer, i.e. It does not converge.
Answer:
D. it does not converge
Jim drank 2/5 of his water bottle and John drank 3/10 of his water bottle. How much did the both boys drink?
Which equation is in point-slope form for the given point and slope?
Answer: y-7 = 4(x+3), choice B
============================================
Point slope form is generally
y-y1 = m(x-x1)
we have m as the slope and (x1,y1) as the point this line goes through. In this case,
m = 4
(x1,y1) = (-3,7) so x1 = -3 and y1 = 7
So,
y-y1 = m(x-x1)
y - 7 = 4(x - (-3))
y - 7 = 4(x + 3)
which is why choice B is the answer
Point-slope form:
y - y₁ = m(x - x₁) "m" is the slope
Since you know:
m = 4
(x₁ , y₁) = (-3,7)
You can plug it into the equation
y - y₁ = m(x - x₁)
y - 7 = 4(x - (-3))
y - 7 = 4(x + 3)
Your answer is the 2nd choice
How do the graphs of these function compare to one another?
how do you do this? solve for b1
[tex]\dfrac{1}{2}h(b_1+b_2)=A\qquad\text{multiply both sides by 2}\\\\h(b_1+b_2)=2A\qquad\text{divide both sides by }\ h\neq0\\\\b_1+b_2=\dfrac{2A}{h}\qquad\text{subtract}\ b_2\ \text{from both sides}\\\\\boxed{b_1=\dfrac{2A}{h}-b_2}\to\boxed{F.}[/tex]
WILL GIVE BRAINLIEST
What is the domain of the function below?
A. -5 ≤ y ≤ 2
B. -3 ≤ x ≤ 3
C. -2 ≤ y ≤ -2
D. -5 ≤ x ≤ 6
Answer:
D
Step-by-step explanation:
The domain is the set of all x-values. We know the domain by finding the start and stop point of the function on the x-axis. The domain is all the values in between if the function is continuous without any breaks.
Here the function begins at x=-5 and ends at x=6.
We write the domain as [tex]-5\leq x\leq 6[/tex].
Answer is D.
Find the next two terms of the sequence: 80, ?40, 20, ?10, ______, ______. A) 20, 40 B) 1 2 , 1 4 C) 5, ? 5 2 D) ?30, ?40
Answer:
5, 5/2
Step-by-step explanation:
80, 40, 20, 10, ______,
80/40 = 2
40/20 = 2
We are dividing by 2 each time
10/2 =5 so the next term is 5
5/2 = 5/2
A rectangular field is 115 yards long and 75 yards wide.
Give the length and width of another rectangular field that has the same perimeter but a smaller area
Final answer:
To create a rectangular field with the same perimeter but a smaller area than the original 115 yards by 75 yards field, one could have a field that is 130 yards long and 60 yards wide.
Explanation:
To find the length and width of another rectangular field that has the same perimeter as a 115 yards by 75 yards field but a smaller area, we first calculate the perimeter of the original field:
Perimeter = 2(length + width) = 2(115 yd + 75 yd) = 2(190 yd) = 380 yd
To have a smaller area, the new field cannot be a square (which would maximize the area) and the sides need to have a greater difference in their measurements while still adding up to half of the original perimeter:
Let's assume the new length is 130 yards, to find the new width: 380 yd / 2 - 130 yd = 60 yd
Therefore, a new rectangular field could be 130 yards long and 60 yards wide.
Need help fast please
Answer:
13. The slope for a line that is perpendicular to the line y=4x+8 is -1/4
14. The slope for a line that is perpendicular to the line x=-6 is 0
15. The equation for a line that is perpendicular to the line 8x-4y=12 and passes through the origin is y = - (1/2) x
16. The equation for a line that is perpendicular to the line y=-(1/3)x and passes through the point (0,-10) is y=3x-10
Step-by-step explanation:
13. What is the slope for a line that is perpendicular to the line y=4x+8?
y=mx+b
Comparing with the form slope-intercept, the slope of the given line is the coefficient of x, then the slope of the given line is 4.
A line perpendicular to y=4x+8 must have a slope opposite and inverse, then:
Slope of the perpendicular = - 1/4
14. What is the slope for a line that is perpendicular to the line x=-6?
The line x=-6 is a vertical line. A line perpendicular to the line x=-6 must be a horizontal line (Angle=0°), then:
Slope of the perpendicular = tan Angle = tan 0° = 0
15. Write the equation for a line that is perpendicular to the line 8x-4y=12 and passes through the origin.
8x-4y=12
Isolating y: Subtracting 8x both sides of the equation:
8x-4y-8x=12-8x
-4y=-8x+12
Dividing all the terms by -4:
-4y/(-4)= -8x/(-4)+12/(-4)
y=2x-3
The slope of the given line is 2
The slope of the perpendicular is m=-1/2 (opposite and inverse to the slope of the given line)
The perpendicular passes through the origin:
P1=(0,0)=(x1,y1)→x1=0, y1=0
Using the equation point - slope:
y-y1=m(x-x1)
Replacing the known values:
y-0=(-1/2)(x-0)
y=(-1/2)x
16. Write an equation for a line that is perpendicular to the line y=-(1/3)x and passes through the point (0,-10).
The slope of the given line is -(1/3)
The slope of the perpendicular is m=3/1→m=3 (opposite and inverse to the slope of the given line)
The perpendicular passes through the point:
P1=(0,-10)=(x1,y1)→x1=0, y1=-10
Using the equation point - slope:
y-y1=m(x-x1)
Replacing the known values:
y-(-10)=3(x-0)
y+10=3x
Isolating y: Subtracting 10 both sides of the equation:
y+10-10=3x-10
y=3x-10
An airplane flies at a speed of 474 miles per hour. How many miles does the plane fly on 5 hours?
Greetings!
Answer:
2370 miles can be travelled
Step-by-step explanation:
If 474 miles can be travelled in 1 hour, this means that to find the total amount in 5 hours we can simply multiply this value by 474:
474 * 5 = 2370
So it can travel 2370 miles!
HELPP! Calculate S22 for the arithmetic sequence in which a12=2.4 and the common difference is d=3.4
Answer:
S22 for the arithmetic sequence is:
First option: 15.4
Step-by-step explanation:
a12=2.4
d=3.4
S22=?
Sn=(a1+an)n/2
n=22
S22=(a1+a22)22/2
S22=(a1+a22)11
ak=aj+(k-j)d
a12=a1+(12-1)d
2.4=a1+11(3.4)
2.4=a1+37.4
Solving for a1: Subtracting 37.4 both sides of the equation:
2.4-37.4=a1+37.4-37.4
Subtracting:
-35=a1
a1=-35
a22=a12+(22-12)d
a22=2.4+10(3.4)
a22=2.4+34
a22=36.4
S22=(a1+a22)11
S22=(-35+36.4)11
S22=(1.4)11
S22=15.4
What is the growth factor of the following example? Assume time is measured in the units given.
Water usage is increasing by 3% per year.
Answer:
The growth factor is option c. 1.03 per year
Step-by-step explanation:
Growth rate: r = 3% = 3 / 100 → r = 0.03
Growth factor: b = 1+r
Replacing r by 0.03 in the formula above:
b = 1+0.03
b = 1.03
Answer: The growth factor is 1.03 per year
Algebraic Expressions Write each phrase as an algebraic expression. 5 subtracted from y
On Friday, 537 people attended a play. On Saturday, 812 people attended the same play. About how many more people attended the play on Saturday than on Friday? Hiw did u estimate? Show your work
Answer:
275
Step-by-step explanation:
To find out how many more, we find the difference between the two values. We use the math operation subtraction to find the difference.
Friday had 537 and Saturday had 812. We subtract 812-537= 275.
275 more people attended Saturday than Friday.
The value of x is 12.
Answer: False
Step-by-step explanation:
45-45-90 is a special triangle because there is a specific relationship between the legs and the hypotenuse: a - a - a√2
So, If the leg of the 45-45-90 triangle is 12, then the hypotenuse (labeled as x in your drawing) will be 12√2.