The required function is f(m) = -53.33m + 360, where f(m) represents the number of frogs in the lake after m months.
What is function ?
In mathematics, a function is a rule that assigns a unique output or "dependent variable" to each input or "independent variable" in a set. It is a relation between a set of inputs and a set of possible outputs, with the property that each input is related to exactly one output.
Based on the given graph, it appears that the number of frogs in the lake is decreasing linearly over time. We can use the slope-intercept form of a linear equation to represent this relationship:
y = mx + b
where y is the number of frogs, x is the time in months, m is the slope (or rate of decrease) of the line, and b is the initial number of frogs at time zero.
To find the slope of the line, we can use the two points (0, 360) and (3, 200) from the graph:
m = (change in y) / (change in x) = (200 - 360) / (3 - 0) = -53.33
So the slope of the line is -53.33, which means that the number of frogs is decreasing by an average of 53.33 per month.
To find the initial number of frogs, we can use the y-intercept of the line, which appears to be around 360 on the graph. So we have:
b = 360
Putting it all together, the function that represents the number of frogs in the lake after m months is:
f(m) = -53.33m + 360
Therefore, the required function is f(m) = -53.33m + 360, where f(m) represents the number of frogs in the lake after m months.
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PLEASE HELP SUPER CONFUSED
Answer:
4.5
Step-by-step explanation:
6+2=8
8/2=4
1.5x4=6
6-1.5=4.5
The ratio of the number of male and female children in the colony is 6:4:3 respectively and the minimum number of members in the colony is 200, then what will be the minimum number of children in the colony?
The minimum number of children in the colony is 47 given the ratio 6 : 4 : 3
Calculating the minimum number of children in the colony?Given that
Ratio = 6 : 4 : 3
We are given the ratio of male to female to children as 6:4:3.
Let the common ratio be x, so the actual number of males, females, and children respectively will be 6x, 4x, and 3x.
Total number of members in the colony = 6x + 4x + 3x = 13x.
We are given that the minimum number of members in the colony is 200.
So,
13x = 200
x = 200/13
The actual number of children in the colony 3x is
Children = 3 * (200/13) = 46.15
Children = 46.15
However, since we cannot have a fraction of a child, we round to the nearest whole number.
Therefore, the minimum number of children in the colony is 47.
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Given that f(x) = 3x – 5 g(x) = 2x – 6 and h(x) = x + 4 4 2x
Therefore , the solution of the given problem of function comes out to be f[g(x)] = 3x² - 2 is the formula for the function f[g(x)].
Explain function.The mathematics programme covers a wide range of topics, including mathematics, numbers, but also their subsets, along with building, construction, and the both real and fictional geographic locations. A work covers the connections between different variable elements that all work together to produce the same result. A utility is made up of several distinctive components that, when combined, produce specific results for each input.
Here,
We must first assess g(x) in order to determine f[g(x)], and then we can use the outcome to determine f[g(x)].
=> g(x) = x² + 1
When we substitute g(x) for f(x), we obtain:
=> f[g(x)] = f(x² + 1)
=> 3(x² + 1) - 5 = 3x² + 3 - 5
=> 3x² - 2
Therefore, f[g(x)] = 3x² - 2 is the formula for the function f[g(x)].
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PLS HELP , in need to solve by using substitution and with checks
Answer:
(x, y) = (-1/2, 5/2)
Step-by-step explanation:
To solve this system of equations using substitution, we can use the first equation to express y in terms of x, and then substitute that expression into the second equation to get an equation in terms of x alone.
Starting with the first equation, we have:
y = x + 3
We can now substitute this expression for y into the second equation:
3x + 3y = 6
3x + 3(x + 3) = 6
Simplifying the right-hand side:
3x + 3x + 9 = 6
6x = -3
x = -1/2
Now that we have solved for x, we can substitute this value back into the first equation to find y:
y = x + 3
y = -1/2 + 3
y = 5/2
Therefore, the solution to the system of equations is (x, y) = (-1/2, 5/2).
To check our solution, we can substitute these values back into both equations and verify that they hold true:
y = x + 3
5/2 = -1/2 + 3
5/2 = 5/2
This equation holds true, so the first equation is satisfied by our solution.
3x + 3y = 6
3(-1/2) + 3(5/2) = 6
-3/2 + 15/2 = 6
6 = 6
This equation also holds true, so the second equation is satisfied by our solution as well.
Therefore, we can conclude that our solution is correct.
The equations have two variables, x and y, and we can use substitution to solve for one variable in terms of the other. By substituting the expression for y into the second equation and solving for x, we get the value of x as -1/2. We can then substitute this value back into the first equation to get the value of y, which is 5/2. This means that the solution to the system of equations is (x, y) = (-1/2, 5/2) and we can check that this solution satisfies both equations.
Hope this helps! Sorry if it doesn't. If you need more help, ask me! :]
Find the zeros of the quadratic function f(x) = 2x^2 – 8x + 6
The answer is 3 and 1
Step-by-step explanation:
Given : 2x ^ 2 - 8x + 6 = 0
To Find: x
Solution:
2x ^ 2 - 8x + 6 = 0
2(x ^ 2 - 4x + 3) = 0
x ^ 2 - 4x + 3 = 0
x ^ 2 - x - 3x + 3 = 0
x(x - 1) - 3(x - 1) = 0
(x - 3)(x - 1) = 0
x - 3 = 0
x = 3
x - 1 = 0
x = 1
Thus the solution are 3 and 1
The number of nations participating in the Winter Games competition has been increasing over the
years, as shown in the table Use linear regression to find a linear function that can be used to predict the
number of nations participating x years after 1988. Then predict those years in which more than 100
nations will participate in the games
What is the linear function that can be used for prediction?
Year
1988
1992
1995
1998
2002
2006
Number of Nations
51
58
61
66
71
74
Answer:
Step-by-step explanation:
o find a linear function that can be used to predict the number of nations participating x years after 1988, we can use linear regression analysis. Using a spreadsheet program or calculator with linear regression capabilities, we obtain:
y = 1.9632x + 51.3276
where y is the predicted number of nations participating and x is the number of years after 1988.
To predict those years in which more than 100 nations will participate, we can substitute y = 100 into the linear function and solve for x:
100 = 1.9632x + 51.3276
48.6724 = 1.9632x
x ≈ 24.8
Therefore, more than 100 nations will participate in the Winter Games approximately 24.8 years after 1988, or in the year 2013.
try to answer all the questions below!
Part 1
a. In function notation, we have that R(3) > D(3)
b. In function notation, we have that R(0) - D(0) = 25
Part 2
The drone is 20 feet above the ground when the rocket hit the ground.
Part 3
a. R(t) = D(t) when the graphs intercept at t = 4.75 s
b. It tell us that the drone and rocket are at the same height at that time.
What is a function notation?A function notation is the representation of a statement as a mathematical equation using symbols.
Part 1
a. To write the statement at 3 seconds the toy rocket is higher than the drone in function notation.
Since
R(t) represents height of toy rocket and D(t) represents height of drone.At t = 3, the toy rocket is higher than the drone.
So, in function notation, we have that R(3) > D(3)
b. To write the statement at the start the toy rocket is 25 feet above the drone.
Since R(t) represents height of toy rocket and D(t) represents height of drone.At the start t = 0, and the toy rocket is 25 feet above the drone.
So, in function notation, we have that R(0) - D(0) = 25
Part 2.
To find the height of the drone when the rocket hit the ground,
Since
R(t) represents height of toy rocket and D(t) represents height of drone.We find R(t) = 0 and find the time t, where it intercepts the height of the drone.
So, from the graph R(t) = 0 at t = 5. And at t = 5, D(t) = 20
So, the drone is 20 feet above the ground when the rocket hit the ground.
Part 3.
a. The value of t at which R(t) = D(t) is where the graphs intercept.
The graphs intercept at t = 4.75 s
b. Since R(t) = D(t) at t = 4.75 s, it tell us that the drone and rocket are at the same height at that time.
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1 1/4Cups of sugar to make 20 cookies. How many cups to make 16 cookies
To make 16 cookies it need 1 cup of sugar.
What are problem solving questions?Problem solving questions are questions that require critical thinking and analytical skills to solve a particular issue or challenge, often in a systematic and logical way.
To make 1 cookie, you need 1 1/4 cups of sugar ÷ 20 cookies = 1/16 cups of sugar.
To make 16 cookies, you need 16 cookies x 1/16 cups of sugar per cookie = 1 cup of sugar.
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Simplify by expressing fractional exponents instead of radicals. ab
The simplification by expressing fractional exponents instead of radicals is a^1/2. b^1/2 or (ab)^(1/2)
How can the simplification be done?The expression in radical form was been provided as √ab and this is been expected from use to express it ionform of fractional exponent form.
√ab
note that;
√a = a^1/2
√ab = √a * √b
By utilixzing the properties of square root function, then we can have the expressions as ;
√ab = √a √b
√ab = a^1/2 * b^1/2
√ab =a^1/2. b^1/2
=a^1/2. b^1/2 or (ab)^(1/2)
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The area of a circle increases at a rate of 5 cm²/s. a. How fast is the radius changing when the radius is 3 cm? b. How fast is the radius changing when the circumference is 1 cm?
Answer:
1. the radius is increasing at a rate of approximately 0.265 cm/s when the radius is 3 cm
2. the radius is increasing at a rate of approximately 0.159 cm/s when the circumference is 1 cm
Step-by-step explanation:
We can use the formulas for the area and circumference of a circle to solve these problems:
a. To find how fast the radius is changing when the radius is 3 cm, we can use the formula for the area of a circle:
A = πr^2
Taking the derivative of both sides with respect to time, t, we get:
dA/dt = 2πr dr/dt
where dr/dt is the rate of change of the radius.
We know that dA/dt = 5 cm²/s, and when the radius is 3 cm, we can plug in these values to solve for dr/dt:
5 = 2π(3) dr/dt
dr/dt = 5/(6π) cm/s ≈ 0.265 cm/s
Therefore, the radius is increasing at a rate of approximately 0.265 cm/s when the radius is 3 cm.
b. To find how fast the radius is changing when the circumference is 1 cm, we can use the formula for the circumference of a circle:
C = 2πr
Taking the derivative of both sides with respect to time, t, we get:
dC/dt = 2π dr/dt
where dr/dt is the rate of change of the radius.
We know that when the circumference is 1 cm, C = 1 cm, so we can plug in these values to solve for dr/dt:
1 = 2π dr/dt
dr/dt = 1/(2π) cm/s ≈ 0.159 cm/s
Therefore, the radius is increasing at a rate of approximately 0.159 cm/s when the circumference is 1 cm.
Marley's car gets 25 miles per gallon, and they expect to drive 1350 miles for their next vacation. If the average price of gasoline is $2.549, how much should they budget for gas for the trip?
A. $63.73
B. $54
C. $137.65
D. $337.50
May any kind soul teach me on how to do this primary six math question .
The unknown length in the rectangle is ? = 8cm
How to find the unknown length?We we want to find the unknown length.
First, if we look just at the pink and yellow rectangles we can see that these have a combined area of:
A = 12cm² + 18cm²
A = 30cm²
And we know that one side measures 6cm, then the length of AD is:
AD*6cm = 30cm²
AD = 30cm²/6cm
AD = 5cm
Now, the yellow and purple rectangles have a combined area of:
A = 18cm² + 22cm² = 40cm²
Then we can write:
?*AD = 40cm²
?*5cm = 40cm²
? = 40cm²/5cm
? = 8cm
That is the unknown lenght.
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A jar contains 7 red marbles, 9 green marbles, and 8 blue marbles. What is the probability that you draw a red marble? = What is the probability that you choose exactly one of each color, if you pick 3 from the jar?= What is the probability that you draw 5 green marbles in a row if you do not replace the marbles after each draw?
1. The prοbability that yοu draw a red marble 7/24
2. The prοbability οf chοοsing exactly οne οf each cοlοr is 63/253
3. The prοbability that yοu draw 5 green marbles in a rοw is 0.0057
What is prοbability?Prοbability is a branch οf mathematics in which the chances οf experiments οccurring are calculated. It is by means οf a prοbability, fοr example, that we can knοw frοm the chance οf getting heads οr tails in the launch οf a cοin tο the chance οf errοr in research.
1. The prοbability οf drawing a red marble is given by the number οf red marbles divided by the tοtal number οf marbles in the jar:
P(red) = 7/(7+9+8) = 7/24
2. The prοbability οf chοοsing exactly οne οf each cοlοr, if yοu pick 3 marbles frοm the jar, can be fοund using the hypergeοmetric distributiοn. The tοtal number οf ways tο chοοse 3 marbles frοm 24 is:
C(24,3) = 2024
The number οf ways tο chοοse οne οf each cοlοr is:
C(7,1) * C(9,1) * C(8,1) = 798 = 504
Therefοre, the prοbability οf chοοsing exactly οne οf each cοlοr is:
P(1 οf each cοlοr) = 504/2024 = 63/253
3. The prοbability οf drawing 5 green marbles in a rοw withοut replacement can be fοund as fοllοws. Therefοre, the prοbability can be calculated as:
P(5 green in a rοw) = (9/24) * (8/23) * (7/22) * (6/21) * (5/20) = 0.0057 (rοunded tο fοur decimal places)
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Find the area of the triangle
Answer:
i don’t know sorry
Step-by-step explanation:
nothing
What are algebraic expressions that are equivalent to 7x-2?
The algebraic expressions that are equivalent to 7x-2 are -2 + 7x, 14x/2 - 4/2 and 5x + 2x - 2
How to determine the equivalent expressionsThe expression in the question is given as
7x - 2
The above expression is linear
Here are some algebraic expressions that are equivalent to 7x-2:
14x/2 - 4/2
When evaluated, we have
14x/2 - 4/2 = 7x - 2
5x + 2x - 2:
We can break up 7x into 5x and 2x, then combine 5x and 2x to get 7x.
So, we have
5x + 2x - 2 = 7x - 2
Hence, the expressions are 14x/2 - 4/2 and 5x + 2x - 2
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Equation of the hyperbola?
The equation of the hyperbola x²/36 = 1. The equation of a horizontal line passing through (±6,0).
Describe Hyperbola?The resulting curve of a hyperbola consists of two separate branches that are mirror images of each other, and they are characterized by their asymptotes, which are straight lines that the curve approaches but never touches. The asymptotes intersect at the center of the hyperbola, which is also the point where the two branches are closest together.
Since the foci of the hyperbola are at (0,-6) and (0,6), and the transverse axis is vertical, the center of the hyperbola is at the midpoint of the foci, which is the point (0,0).
Also, since the asymptote of the hyperbola is the line y = -x, we know that the slopes of the asymptotes are ±1, and since the transverse axis is vertical, the slope of the transverse axis is 0. Therefore, we have a hyperbola with center (0,0), vertical transverse axis, and asymptotes y = x and y = -x.
The standard form of the equation for a hyperbola with center (0,0), transverse axis of length 2a, and asymptotes y = ±(a/x) is:
x²/a² - y²/b² = 1
where b² = a² - c², where c is the distance from the center to each focus.
In this case, since the distance between the foci is 12, we have:
c = 6
Also, since the transverse axis is vertical and the distance from the center to each vertex is a, we have:
a = distance from center to vertex = 6
Therefore, we have:
b² = a² - c² = 36 - 36 = 0
So the equation of the hyperbola is:
x²/36 - y²/0 = 1
Simplifying, we get:
x²/36 = 1
or
x² = 36
This is the equation of a horizontal line passing through (±6,0).
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PLEASE HELP WILL GIVE THE BRAINLISET AND 60 PTS !!!
Answer:C
Step-by-step explanation:
you see x=-3 is match for y=2
Answer:
2
Step-by-step explanation:
Look for where the line crosses x=-3. It crosses at y=2. so the answer is the third, 2
The tallest tower built before the era of television masts was the Eiffel Tower which was completed on March 31, 1889. If you stand at a distance of 80 ft from a point directly beneath the center of the tower, you will find the angle of elevation to the top of the tower to be a staggering 85°. How tall is the Eiffel Tower? Round the answer to the nearest tenth.
Answer:
We can use trigonometry to solve the problem. Let's call the height of the tower "h". Then we can use the tangent function to find "h":
tan(85°) = h/80
Multiplying both sides by 80, we get:
h = 80 tan(85°) ≈ 884.2 ft
Rounding to the nearest tenth, the height of the Eiffel Tower is approximately 884.2 ft.
Step-by-step explanation:
Find the value of the unknown.
162000=750×6^y
Answer:
[tex]\boxed{\textsf{y = 3}}[/tex]
Step-by-step explanation:
[tex]\textsf{We are asked to solve for y.}[/tex]
[tex]\textsf{We should begin by \underline{simplifying} the equation.}[/tex]
[tex]\textsf{First, begin by dividing 750 from \underline{both sides} of the equation.}\\[/tex]
[tex]\Large\underline{\textsf{Divide:}}[/tex]
[tex]\mathtt{\frac{162000}{750} = \frac{750 \times 6^y}{750} .}[/tex]
[tex]\mathtt{216=6^y}[/tex]
[tex]\Large\underline{\textsf{Continue Dividing By 6:}}[/tex]
[tex]\mathtt{\frac{216}{6} = 36 (^{y-1})}[/tex]
[tex]\mathtt{\frac{36}{6} = 6 (^{y-2} )}[/tex]
[tex]\Large\underline{\textsf{Identify y:}}[/tex]
[tex]\mathtt{6^1 = 6. \ 6^{1+2} = 216. \ (6^3)}[/tex]
[tex]\boxed{\textsf{y = 3}}[/tex]
If the angle -270° rotates 180° counter-clockwise, what is the new measurement of the co-terminal angle?
___°
Starting with the angle of -270°, adding 360° gives us 90° (which is a coterminal angle with -270°).
If we rotate 180° counter-clockwise from 90°, we end up at the angle of 270°. Therefore, the new measurement of the coterminal angle is 270°.
I hope this helps :)
A bag contains several marbles 7 of which are blue 18 are yellow and 12 are green
If 3 marbles are chosen without replacement what is the probability that they are all yellow
Answer:
The answer would be 8.1% (8.108% if you need the exact percentage.)
Step-by-step explanation:
All you would need to do is add 7 + 18 + 12 which = 37, and divide 3/37, which in percentage that gives you 8.1%. (8.108% if you need exact percentage.)
PLEASE ANSWER ASAP PLEASE
value from least to greatest is
A<B<C<D
Define fractionA fraction is a numerical quantity that represents a part of a whole or a ratio of two quantities. It is expressed in the form of one number, called the numerator, divided by another number, called the denominator, and is typically written as a/b, where "a" is the numerator and "b" is the denominator.
Part a)
Let 5.7 (bar on 7) be x
x=5.777777... .....(1)
Multiply the equation by 10,
10x=57.77777... ....(2)
(2)-(1),
9x=52
x=52/9=5.777777778
Partb)
Let 5.75 (bar on 75) be x
x=5.75757575... .....(1)
Multiply the equation by 100,
100x=575.75757575.. ....(2)
(2)-(1),
99x=569.99
x=190/33=5.757575758
Part c)
Let 5.75 (bar on 5) be x
x=5.7555555... .....(1)
Multiply the equation by 10,
10x=57.55555555... ....(2)
Multiply the equation by 10,
100x=575.5555.. ....(3)
(3)-(2),
90x=52
x=259/45=5.755555556
Hence, value from least to greatest is
A<B<C<D
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The answer of the given question based on the ordering from the least to greatest the answer is the correct answer is (A) DBCA.
What is Decimal system?The decimal system is a number system that uses the base 10, meaning that there are 10 digits (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) that can be used to represent any quantity. This system is also called the base-10 numeral system, denary system, or Hindu-Arabic numeral system.
In the decimal system, each digit represents a different power of 10. For example, in the number 357, the 7 represents the units, 5 represents the tens, and 3 represents the hundreds.
This system is widely used in everyday life, in which numbers are expressed in decimal form, such as money, time, and measurements. It is also used in mathematics, science, and engineering as the standard way of expressing numbers.
The correct order from least to greatest will be :
A. 5.7
D. 5.75
B. 5.75
C. 5.75
Therefore, the correct answer is (A) DBCA.
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-4x = 64
How should you show your work to isolate the variable and solve the equation?
Answer:
-16
Step-by-step explanation:
1. multiply the minus sign on both sides now we have 4x=-64
2. divide 4 on both sides
You're done! x=-16
The solution to a system of equations is (5.-19). Choose two equations that might make up the system.
Oy=2x-23
Oy=x-17
Oy=-7x+16
Dy=-21-9
Oy=-3x-6
Answer: The solution to a system of equations in two variables represents the values of the variables that satisfy both equations simultaneously.
To check which equations might make up the system with a solution of (5,-19), we can substitute x = 5 and y = -19 into each equation and see if they are both satisfied.
Substituting x = 5 and y = -19 into the first equation, we get:
y = 2x - 23
-19 = 2(5) - 23
-19 = -13
This is not true, so the first equation is not part of the system.
Substituting x = 5 and y = -19 into the second equation, we get:
y = x - 17
-19 = 5 - 17
-19 = -12
This is not true, so the second equation is not part of the system.
Substituting x = 5 and y = -19 into the third equation, we get:
y = -7x + 16
-19 = -7(5) + 16
-19 = -19
This is true, so the third equation is one of the equations in the system.
Substituting x = 5 and y = -19 into the fourth equation, we get:
y = -21 - 9x
-19 = -21 - 9(5)
-19 = -64
This is not true, so the fourth equation is not part of the system.
Substituting x = 5 and y = -19 into the fifth equation, we get:
y = -3x - 6
-19 = -3(5) - 6
-19 = -21
This is not true, so the fifth equation is not part of the system.
Therefore, one possible system of equations with a solution of (5,-19) is:
y = -7x + 16
We would need another equation to form a complete system with a unique solution.
Step-by-step explanation:
3. Karem drives his car 96 miles in 1 ½ hours. If Karem continues to drive at this rate, how many miles will he travel in 2 ½ hours?
Answer:
160 miles
Step-by-step explanation:
To find the unit rate, we want to know how many miles can we go in i hour.
96 ÷ 1 [tex]\frac{1}{2}[/tex]
1 [tex]\frac{1}{2}[/tex] can be written
[tex]\frac{2}{2}[/tex] + [tex]\frac{1}{2}[/tex] = [tex]\frac{3}{2}[/tex] 1 is the same as [tex]\frac{2}{2}[/tex]
Now we have
96 ÷ [tex]\frac{3}{2}[/tex] if we get a common denominator, in this case 2, we can just divide a cross.
96 is equal to [tex]\frac{192}{2}[/tex] ([tex]\frac{96}{1}[/tex] x [tex]\frac{2}{2}[/tex])
Now we have
[tex]\frac{192}{2}[/tex] ÷ [tex]\frac{3}{2}[/tex] = [tex]\frac{64}{1}[/tex] = 64
We now know the unit rate is 64 miles in one hour.
The distance in 2 [tex]\frac{1}{2}[/tex] hours would be
64 + 64 + [tex]\frac{1}{2}[/tex](64)
64 + 64 + 32
160
Helping in the name of Jesus.
Consider a circle whose equation is x2 + y2 – 2x – 8 = 0. Which statements are true? Select three options. The radius of the circle is 3 units. The center of the circle lies on the x-axis. The center of the circle lies on the y-axis. The standard form of the equation is (x – 1)² + y² = 3. The radius of this circle is the same as the radius of the circle whose equation is x² + y² = 9.
The center of the circle lies on the x-axis.
The radius of the circle is 3 units.
The radius of this circle is the same as the radius of the circle whose equation is x² + y² = 9.
The standard equation of a circle is expressed as: Centre is (-g, -f)radius = √g²+f²-CGiven a circle whose equation is Get the centre of the circle2gx = -2x2g = -2g = -1Similarly, 2fy = 0f = 0Centre = (-(-1), 0) = (1, 0)This shows that the center of the circle lies on the x-axisr = radius = √g²+f²-Cradius = √1²+0²-(-8)radius =√9 = 3 unitsThe radius of the circle is 3 units.
For the circle x² + y² = 9, the radius is expressed as:r² = 9r = 3 units Hence the radius of this circle is the same as the radius of the circle whose equation is x² + y² = 9.
Convert the following to logarithmic form:
a^3 = y
Choose one:
a. 0 = loga y
b. 3 = loga y
c. a = log3 y
d. - 3 = loga y
The logarithmic form of the equation a^3 = y is log_a(y) = 3
What is a logarithmic equationA logarithmic equation is an equation that involves the logarithm of one or more variables.
How to convert the expression to a logarithmic formFrom the question, we have the following parameters that can be used in our computation:
a^3 = y
Take the logarithm of both sides of the equation
So, we have
3log(a) = log(y)
Divide both side by log(a)
So, we have
3 = log_a(y)
Rewrte as
log_a(y) = 3
Hence, the equation is (b) 3 = log_a(y)
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3^-2 find the equivalent expression from these choices
The fourth expression 3⁻⁷/3⁻⁵ is equivalent to 3⁻².
What exactly are expressions?
In mathematics, an expression is a combination of numbers, variables, operators, and functions that are combined in a particular way to represent a mathematical value or relationship. Expressions can be simple or complex, and they can involve any number of mathematical operations.
Expressions can be used to represent many mathematical concepts:
Arithmetic operations: addition, subtraction, multiplication, and divisionExponents and logarithmsTrigonometric functions: sine, cosine, and tangentAlgebraic equations: variables and constantsCalculus: derivatives, integrals, and limitsNow,
To find the equivalent expression of 3⁻², we can simplify each of the given choices using the rules of exponents:
1. 3⁻²/3³ = 1/3²⁺³ = 1/3⁵
2. 3⁻⁷/3⁵ = 3⁻⁷⁻⁵ = 3⁻¹²
3. 3³/3⁻¹² = 3³⁺¹² = 3¹⁵
4. 3⁻⁷/3⁻⁵ = 3⁻⁷⁺⁵ = 3⁻²
Out of the given choices, the fourth choice 3⁻⁷/3⁻⁵ is equivalent to 3⁻².
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The U.S. population in 1910 was 92 million people. In 1990, the population was 280 million. Create both a linear and an exponential model of the population from 1910 to 1990, with projected data points at least every 20 years, starting with 1910 as year 0. Include both an equation and a graph in your answer.
(Disregard what I wrote - My eraser didn't work well)
The linear model is Population = 92 + 2.35x and the exponential model is Population= 92(1.0187)ˣ.
What distinguishes an exponential model from a linear model?An exponential model implies that the change in y is proportionate to its present value, whereas a linear model assumes that the change in y is constant for each unit change in the explanatory variable (x). To put it another way, an exponential model assumes a varying rate of change, whereas a linear model implies a constant rate of change.
Let us suppose number of years = x.
Given that, the U.S. population in 1910 was 92 million people. In 1990, the population was 280 million.
From 1910 to 1990 is a period of 80 years, the increase in population over this period is 280 million - 92 million = 188 million.
Therefore, the average annual increase in population is 188 million / 80 years = 2.35 million.
Thus, the linear model is:
Population = 92 + 2.35x
The exponential model is:
The population grew from 92 million in 1910 to 280 million in 1990, a factor of 280/92 = 3.0435.
The period of growth is 80 years, so the annual growth rate can be calculated as:
rate = (3.0435)^(1/80) - 1 = 0.0187 or 1.87%
Population= 92(1.0187)ˣ
Hence, the linear model is Population = 92 + 2.35x and the exponential model is Population= 92(1.0187)ˣ.
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Lush Gardens Co. bought a new truck for $54,000. It paid $5,940 of this amount as a down payment and financed the balance at 4.46% compounded semi-annually. If the company makes payments of $1,600 at the end of every month, how long will it take to settle the loan?
0
years
0
months?
Express the answer in years and months, rounded to the next payment period
It will take Lush Gardens Co. 3 years and 3 months to settle the loan if it makes payments of $1,600 at the end of every month.
To find how long will it take to settle the loan?First, we need to find the amount of the loan.
Amount of loan = Total cost of the truck - Down payment
Amount of loan = $54,000 - $5,940 = $48,060
Next, we need to find the monthly interest rate. Since the interest is compounded semi-annually, we first need to find the semi-annual interest rate:
Semi-annual interest rate = 4.46% / 2 = 2.23%
Then, we can find the monthly interest rate using the formula:
(1 + Monthly interest rate)^12 = 1 + Semi-annual interest rate(1 + Monthly interest rate) = (1 + Semi-annual interest rate)^(1/12)Monthly interest rate = (1 + Semi-annual interest rate)^(1/12) - 1Plugging in the numbers:
Monthly interest rate = (1 + 0.0223)^(1/12) - 1 = 0.001845
Now we can use the formula for the loan payment:
Loan payment = Monthly interest rate * Loan amount / (1 - (1 + Monthly interest rate)^(-n))
Where n is the number of months.
Plugging in the numbers:
$1,600 = 0.001845 * $48,060 / (1 - (1 + 0.001845)^(-n))
Solving for n using a financial calculator :
n = 38.98 months
Rounding up to the next payment period (which is 39 months), we get:
n = 3 years, 3 months
Therefore, it will take Lush Gardens Co. 3 years and 3 months to settle the loan if it makes payments of $1,600 at the end of every month.
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