Answer:
[tex]f(x)=4(3)^x[/tex]
Step-by-step explanation:
The general equation for an exponential function is:
[tex]\boxed{f(x) = ab^x}[/tex]
where:
a is the initial value or y-intercept.b is the base or growth factor.To find the values of a and b that satisfy the given conditions, we can use the two points (1, 12) and (3, 108) to form a system of equations:
[tex]\begin{cases}12 = ab^1\\108 = ab^3\end{cases}[/tex]
Divide the second equation by the first equation to eliminate a:
[tex]\dfrac{ab^3}{ab} = \dfrac{108}{12}[/tex]
[tex]b^2=9[/tex]
[tex]\sqrt{b^2}=\sqrt{9}[/tex]
[tex]b=3[/tex]
Substitute the found value of b into the first equation and solve for a:
[tex]12&=3a[/tex]
[tex]\dfrac{12}{3}=\dfrac{3a}{3}[/tex]
[tex]4=a[/tex]
[tex]a=4[/tex]
Therefore, the equation of the exponential function that passes through the points (1, 12) and (3, 108) is:
[tex]\boxed{f(x) = 4(3)^x}[/tex]
Answer:
y=4(3^x).Step-by-step explanation:
To find the equation of an exponential function passing through the given points (1,12) and (3,108),
we can use the standard exponential form y=a(b^x). We know that when x=1, y=12,
so we can substitute these values into the equation to find a.
So 12 = a(b^1). Similarly, when x=3, y=108, so 108 = a(b^3). We can divide the second equation by the first to eliminate a and get (108/12) = b^2, or 9 = b^2. Thus, b=3 (taking only the positive root). We can now substitute this value of b into either equation to find a. Using the first equation, we get 12 = a(3^1), so a=4. Therefore, the exponential function passing through the given points is y=4(3^x).
Two mechanics worked on a car. The first mechanic worked for 10 hours, and the second mechanic worked for 15 hours. Together they charged a total of 1650 . What was the rate charged per hour by each mechanic if the sum of the two rates was 135 per hour?
Rates of first and second mechanics are 75 and 60 per hour respectively.
What does an equation mean?A mathematical statement or relation between two or more numeric values, variable and mathematical operations via equal sign is called an Equation.
How to solve equations in Elimination Method?Elimination method is a kind of method to solve simultaneous equations in two variables. In elimination method we compare coefficient of one of two variables and in next step we eliminate that variable using addition or subtraction and form another relation with only one variable and find the value for that one variable and then substituting that value initial equation we can get the value of another variable.
Let the rate of first mechanic and second mechanic be x and y per hour respectively.
Sum of the two rates is 135 per hour. So the suitable equation will be,
x+y = 135 ............... (i)
Since they charged together 1650 after working 10 hours by first and 15 hours by second mechanics. So the suitable equation this time will be,
10x+15y = 1650
5(2x+3y) = 1650
2x+3y = 1650/5
2x+3y = 330 ............... (ii)
Multiplying 2 with equation (i) and then subtracting it from equation (ii) we get,
(2x+3y) - 2(x+y) = 330 - 2*135
2x+3y-2x-2y = 330-270
y = 60
Substituting y=60 in equation (i) we get,
x+60 = 135
x = 135-60 = 75
Hence, the rates charged per hour are 75 and 60 per hour respectively.
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What’s the volume of this? pls do step by step cus it’s for my homework and I have to show work
solve 3/4 (5x- 3) + 8 = 17. show your work.
i just dont want to do this
Answer:
B
[tex]6^{-7}[/tex] is equivalent to 1/279936 which is basically [tex]\frac{1}{6^{7}}[/tex]
How does a person solve this
Mr. Rose is remodeling his house by adding a room to one side, as shown in the diagram below. In order to determine the length of the boards he needs for the roof of the room, he must calculate the distance from point A to point D
The length or of the segment AD or distance from point A to point D is 16.6 ft.
Given that Mr. Rose is remodeling his house by adding a room to one side
In order to determine the length of the boards he needs for the roof of the room, he must calculate the distance from point A to point D
We have to find the distance of AD
Sin 25 = 7/x
x is the distance of AD
x=7/sin 25
x=16.6 ft
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Note: Enter your answer and show all the steps that you use to solve this problem in the space provided.
Find the area of the parallelogram.
check the picture please.
Answer:
[tex]192[/tex] [tex]cm^2[/tex]
Step-by-step explanation:
Recall that the formula for the area of a parallelogram is:
[tex]A=bh[/tex]
Where b is the length of the base, and h is the length of the height.
We are given that the base is 24 cm.
So, b=24.
We are also given that the height is 8 cm.
So, h=8.
Let's plug these values into the formula:
[tex]A=bh=\\A=(24)(8)=\\A=192[/tex]
Since area is a two-dimensional measure, we need to label our answer using units squared.
Thus, the area is [tex]192[/tex] [tex]cm^2[/tex]
Area of a parallelogram => [tex]A=bh[/tex]
Where, b=24 cm and h=8 cm
Plug in the known values and solve.
[tex]\Longrightarrow A=bh \Longrightarrow A=(24 \ cm)(8 \ cm) \Longrightarrow A= \boxed{192 \ cm^2}[/tex]
Jungle John participates in a reality TV show that features fitness and agility competitions. In order to beat the competitors, John needs to swing from tree A to tree B, completing an arc of at least 60 feet with a central angle of 115∘.
What length of rope (radius) will he need to accomplish this activity?
Round your answer to the nearest foot, and please show your work.
The length of rope Jungle John will he need to accomplish this activity is 30 feet.
How do we calculate?
the formula for arc length is
L = rθ
where L = length of the arc,
r =e radius of the circle,
θ = central angle of the arc in radians.
θ = (115∘/180∘)π = 2.0089 radians
60 feet = r(2.0089 radians)
r = 60 feet / 2.0089 radians
r = 29.85 feet
r is approximated to 30 feet
Therefore, Jungle John will need a rope with a length of at least 30 feet to swing from tree A to tree B and complete an arc of at least 60 feet with a central angle of 115∘.
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Olivia mowed 4 lawns in 16 hours. What was her rate of mowing in hours per lawn?
Answer:
Her rate of mowing lawns per hour is 4 lawns to one hour
Step-by-step explanation:so on a table you divide 4÷4=1 then you do the same thing for the 16÷4=4. So Oliva can mow 4 lawns per hour
Is (7,-6) a solution to this system of equations?
y = 1/7x+ 7
X = 7
yes
no
ASAAP NEED IT RN PLEASE HELP
Thanks :)
The equation of circle in the standard with given center and radius is [tex](x+3)^{2} + (y-4)^2[/tex] = 49.
What is equation of circle?
Every point in a plane that is at a certain distance from the center point forms a circle. In order to go around a curve while maintaining a constant distance from another point, a moving point in a plane must follow a specific path.
The following is the typical form for expressing the equation of a circle:
[tex](x-h)^{2} + (y-k)^2 = r^2[/tex]
Here,
(h, k) represents the center and r is the radius.
We are given that the center is (-3, 4) and the radius is 7.
Now, using the standard form, we get
⇒ [tex](x-(-3))^{2} + (y-4)^2 = 7^2[/tex]
⇒ [tex](x+3)^{2} + (y-4)^2[/tex] = 49
Hence, the equation of circle in the standard with given center and radius is [tex](x+3)^{2} + (y-4)^2[/tex] = 49.
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Since, there are multiple questions so the question answered above is attached below.
Musa got 750 bags of coffee
2011 yield dropped by 30%
2012 rose by 15%
A bag of coffee weighs 55kg and he paid 7900 shillings per tonne
Thereafter the price per tonne increased by 10% find his earnings from coffee hence find his total income for the three years
Answer:
2011: 0.3×750= 225
therefore, 750-225=525
2012: 0.15 × 525= 78.75
therefore, 525 + 78.75= 603. 75
bag of coffee: mass (kg)
1: 55
603.75: x
x= 55 × 603.75
=33206.25 Kg
33206.25÷1000= 332.0625
tonne: shillings
1:7900
332.0625: x
hence, x= 2623293.75 shillings
8690× 332.0625= 2885623. 125 shillings
4. The geometry of a gear tooth is approximated by the following quadratics equation:
A. Determine the height h of the gear tooth
B: determine the area of the gear tooth by integration with respect to x
Using Integration, Area of the curve is 16/3 unit² and the maximum height of the curve is 2 unit.
What exactly is integration?
Integration is a fundamental concept in calculus, which is a branch of mathematics that deals with rates of change and accumulated quantities. Integration is the process of finding the integral of a function, which is the inverse of differentiation.
Now,
For the maximum height that the curve has we have to
put dy/dx=0
d(4x-2x²)/dx=0
4-4x=0
x=4/4
x=1
and when x=1 then y=4*1-2*1²
=4-2
=2
So, the maximum height of the curve is 2 unit.
and for the area
we need to find y.dx for the curve
So,
y.dx=(4x-2x²).dx
=4x+2x²-2x³-2x³/3 for x=0 to x=2
=8+8-16-16/3
=16/3 unit²
Hence, Area of the curve is 16/3 unit².
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What is the simplified form of 7 log3x + 4 log3z – 6 log3y?
The simplified form of 7 log3x + 4 log3z - 6 log3y is log3(x^7 z^4 / y^6).
What is the simplified form of the expressionWe can simplify the expression using the following logarithmic rule:
log a + log b = log ab (product rule)
log a - log b = log(a/b) (quotient rule)
c log a = log a^c (power rule)
Using these rules, we can simplify 7 log3x + 4 log3z - 6 log3y as follows:
7 log3x + 4 log3z - 6 log3y
= log3x^7 + log3z^4 - log3y^6 (using the power rule)
= log3(x^7 z^4 / y^6) (using the quotient rule)
Therefore, the simplified form of 7 log3x + 4 log3z - 6 log3y is log3(x^7 z^4 / y^6).
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A sphere has a volume of approximately 1332 cubic feet.
What is the radius of the sphere?
Round your answer to the nearest tenth if needed.
1
feet
Answer:
[tex] \frac{4}{3} \pi {r}^{3} = 1332[/tex]
[tex]r = \sqrt[3]{ \frac{1332}{ \frac{4}{3}\pi } } = 6.8[/tex]
The radius of this sphere is about 6.8 feet
3 divide 2 5ths in a facrtion
When 3 is divided by 2/5 , the answer is 7.5
Let's compare our whole number and fraction so that we can see the issue we are attempting to tackle.
3 divided by 2/5
In this case, creating a new numerator just requires multiplying the numerator by the full number. After that, the original numerator becomes the new denominator.
3*5/2 = 15/2
Most people prefer to express results in decimals, and all you have to do to do so is divide the numerator by the denominator:
15/2 = 7.5
What is Numerator?The portion written above the horizontal line is referred to as the numerator. Example: 2/5 ,here 2 is numerator.
What is Decimal?A decimal number consists of both a whole number and a fractional number. The numerical value of complete and partially whole amounts is expressed using decimal numbers, which are in between integers.
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The complete question is
what happens when 3 divide by 2/5ths in a facrtion?
Is (3, 5) a solution to this system of equations?
y=5
y = -5/3x + 10
yes
no
The polynomial function f has exactly one positive zero. Approximate the zero correct to two decimal
places.
f(x)=2x-16x³ - 3x² - 8x-2
The positive zero of f is approximately
(Round to two decimal places as needed.)
141
Answer:
We can use a numerical method such as the Newton-Raphson method or the bisection method to approximate the positive zero of the function f(x) = 2x - 16x³ - 3x² - 8x - 2.
Let's use the Newton-Raphson method to approximate the positive zero of f(x). We start by choosing an initial guess x_0, and then compute successive approximations using the formula:
x_(n+1) = x_n - f(x_n) / f'(x_n)
where f'(x) is the derivative of f(x). We continue this process until we get an approximation that is accurate enough for our needs.
Let's choose an initial guess of x_0 = 1.5. Then we have:
f(x) = 2x - 16x³ - 3x² - 8x - 2
f'(x) = 2 - 48x² - 6x - 8
Using these expressions, we can compute successive approximations as follows:
x_1 = x_0 - f(x_0) / f'(x_0) = 1.5 - (-23.375) / (-73) ≈ 1.320
x_2 = x_1 - f(x_1) / f'(x_1) = 1.320 - (-12.608) / (-50.673) ≈ 1.141
x_3 = x_2 - f(x_2) / f'(x_2) = 1.141 - (-5.364) / (-35.883) ≈ 1.067
x_4 = x_3 - f(x_3) / f'(x_3) = 1.067 - (-1.949) / (-31.120) ≈ 1.042
x_5 = x_4 - f(x_4) / f'(x_4) = 1.042 - (-0.361) / (-30.251) ≈ 1.029
x_6 = x_5 - f(x_5) / f'(x_5) = 1.029 - (-0.012) / (-30.055) ≈ 1.028
So the positive zero of f(x) is approximately 1.028, rounded to two decimal places.
If you look up the definition of absolute in the dictionary, you’ll find several definitions. All of them reference totality, completeness, and having no limitations or dependence on anything else. The definition of the absolute value of a number is its magnitude, or distance, from zero. How do these two definitions connect?
A relationship between two definitions can be seen through the idea of magnitude or distance.
How do these two definitions connect?Finding an absolute value of a number requires us to inspect the difference between that figure and zero on a number line; this amount is always positive regardless of whether the initial number was negative or positive. To illustrate, the absolute value of -5 is 5 since there is only five units separating it from zero.
The absolute value primarily entails a totality- recognizing the entirety of someone's peculiarity independent of their direction. It serves as a method of measuring the scope or size of a number without any external parameters or assessment.
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12. Mr. Wright is five times as old as
his daughter, Lilly. Lilly is twice
as old as her brother, Micah. If
Micah is 3 years old, how old is
Mr. Wright?
Based on the given data, the age of Mr. Wright currently is 30 years
If Micah is 3 years old, then Lilly is twice as old, which means that
Lilly is 2 * 3 = 6 years old.
Now we know that Mr. Wright is five times as old as his daughter, Lilly, so Mr. Wright's age is:
5 * 6 = 30 years old.
Therefore, Mr. Wright is 30 years old.
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Answer please and thank you
Answer:
Step-by-step explanation
The answer is 72. 2 x 36 equals 72.
A rectangle has a length of x + 4 cm and a width of 2x − 7 cm.
(a) If the perimeter is 36cm, what is the value of x?
(b) What is the area of this rectangle
Answer:
the value of x is 7
Step-by-step explanation:
First, find the half of the perimeter that is 36 divided by 2 which is 18
Next, write like this
18= l+w( length = width)
which means you have to get the sum of 18
so try sums of 18 randomly like
17 + 1
11 + 7
16+2 etc
but the thing is the value of x must be the same it must not be different so
11 + 7 will be suitable because
11 = 7 + 4
7 = 2 x 7 - 7 = 7
and to find the area which is l x b
11 x 7= 77sq. cm
please mark me the brainliest and a thanks too
Answer:
x=7
Step-by-step explanation:
7+4=11(2)=22
2(7)-7= 14-7= 7(2)=14
22+14=36
The Swamis family went on a vacation. They recorded the number of miles they drove each day. The mileage is shown in the table. Day 1 2 3 4 5 Distance (mi) 248 176 379 263 202
a) The mean of the mileage is 90.4 miles per day
b) The median of the mileage is 255.5 miles
Given data ,
Let the data be represented as A
Now , the value of A is
A = { 248 176 379 263 202 }
And , Mean = (248 + 176 + 379 + 263 + 202) / 5 = 452 / 5 = 90.4 miles per day (rounded to one decimal place)
Therefore, the mean of the data is 90.4 miles per day.
To find the median of the data, we need to first put the distances in order from least to greatest:
176, 202, 248, 263, 379
There are 5 data points, so the median is the middle value.
Median = (248 + 263) / 2 = 511 / 2 = 255.5 miles
Hence , the median of the data is 255.5 miles
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The complete question is attached below :
The Swamis family went on a vacation. They recorded the number of miles they drove each day. The mileage is shown in the table. Day 1 2 3 4 5 Distance (mi) 248 176 379 263 202 What is the mean of the data? ____miles What is the median of the data? ______miles
Is a triangle with sides of 5 meters, 12 meters, and 13 meters a right triangle? Explain/Show your work.
Answer:
[tex]\huge\boxed{\sf Yes.}[/tex]
Step-by-step explanation:
In a right angled triangle, the longest side is called hypotenuse.
So,
Hypotenuse = 13 m
The rest are:
Base = 5 m
Perpendicular = 12 m
Using Pythagoras Theorem to verify:
[tex](Hypotenuse)^2=(Base)^2+(Perpendicular)^2[/tex]
Put the given data
(13)² = (12)² + (5)²
169 = 144 + 25
169 = 169Since, both left hand side and right hand side are equal, this means the triangle is a right-angled triangle.
[tex]\rule[225]{225}{2}[/tex]
La sala de un museo alberga una exposición de insectos.
En total hay 90 insectos y 45 de ellos son mariposas.
¿Qué porcentaje de los insectos son mariposas?
6 A cube-shaped block of cheese has edge lengths of 8 inches. The block of cheese is cut into smaller pieces. Each piece has a volume of 1 cubic inch. How many pieces of cheese will there be? 16 pieces 64 pieces 128 pieces O. 512 pieces
When the block is cut into smaller pieces, 512 pieces of cheese remain.
What is cube and formula of volume of cube?A cube is a solid three-dimensional shape with 6 square faces, 8 vertices and 12 edges. It is also said to be a regular hexahedral.
The volume V of a cube is given by the formula V = a^3, where a = the length of one side of the cube. V = 4 ^ 3 = 64 cubic meters or inches ^ 3. The volume of a cube is 64 cubic meters
The total volume of the cheese block is obtained as follows:
V = edge length³ = 8³ = 512 cubic meters
Since the volume of each piece is 1 cubic inch, the total number of pieces is obtained by dividing the total volume by the volume of each piece:
Number of pieces = V / volume per piece = 512 / 1 = 512 pieces
Therefore, when the block is cut into smaller pieces, 512 pieces of cheese remain.
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Chau paid 13.88 for a -pound bag of shrimp at one store. The following week, he paid 38.99 for a 5.46 -pound bag at another store.
The first bag has a better unit price, at $7.01 per pound compared to $7.13 per pound for the second bag. Based on the unit price, the first bag is a better buy.
Describe Algebra?Algebra is a branch of mathematics that deals with mathematical symbols and the rules for manipulating these symbols. It involves the use of variables, which are represented by letters or other symbols, to represent unknown quantities and to formulate and solve equations and inequalities.
The fundamental operations of algebra include addition, subtraction, multiplication, and division, which are used to manipulate the variables and simplify expressions. Algebra also involves the use of exponents, which are used to represent repeated multiplication, and logarithms, which are used to represent the inverse of exponents.
Algebra is used in a wide range of applications, including science, engineering, economics, and statistics, to model and analyze various phenomena. It is also used as a tool for problem-solving and decision-making, and it provides a powerful framework for reasoning and abstraction in many areas of mathematics and beyond.
To find the unit price, we divide the total price of each bag by its weight:
For the first bag:
Unit price = Total price / Weight = 13.88 / 1.98 = 7.01 dollars per pound
For the second bag:
Unit price = Total price / Weight = 38.99 / 5.46 = 7.13 dollars per pound
Therefore, the first bag has a better unit price, at $7.01 per pound compared to $7.13 per pound for the second bag.
Thus, based on the unit price, the first bag is a better buy.
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The complete question is:
Chau paid 13.88 for a 1.98- pound bag of shrimp at one store. The following week, he paid 38.99 for a 5.46 -pound bag at another store.
Find the unit price for each bag. Then state which bag is better buy based on the unit price.
NO LINKS!! URGENT HELP PLEASE!!!
Express the statement as an inequality Part 4a^2
The correct answer is B) |x| > 6 for the given statement
What is meant by the statement?
A statement is a declarative sentence that can be either true or false. It is a proposition that can be proven using logical reasoning or evidence. Statements are the building blocks of mathematical proofs and are used to establish the truth of mathematical theories and concepts.
According to the given information
The statement “the absolute value of x is greater than 6” can be expressed as an inequality in the following way:
|x| > 6
This means that the distance between x and 0 on the number line is greater than 6 units. In other words, x can be any number that is either less than -6 or greater than 6.
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Employees at a factory receive regular raises. The table below shows how an employee's hourly wage increases based on these regular raises. Which linear equation models the relationship shown in the table? A: y=0.5x+0.8 B: y=1.6x+9.75 C: y=9.75+1.6 D: y=0.8x+9.75
A linear equation that models the relationship shown in the table is: D. y = 0.8x + 9.75
How to determine an equation of this line?In Mathematics and Geometry, the point-slope form of a straight line can be calculated by using the following mathematical equation (formula):
y - y₁ = m(x - x₁)
Where:
x and y represent the data points.m represent the slope.First of all, we would determine the slope of this line;
Slope (m) = (y₂ - y₁)/(x₂ - x₁)
Slope (m) = (10.55 - 9.75)/(1 - 0)
Slope (m) = 0.8/1
Slope (m) = 0.8
At data point (0, 9.75) and a slope of 0.8, a linear equation for this line can be calculated by using the point-slope form as follows:
y - y₁ = m(x - x₁)
y - 9.75 = 0.8(x - 0)
y = 0.8x + 9.75
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Missing information:
The question is incomplete and the complete question is shown in the attached picture.
m^2-6m +9
in factored form?
Answer:
(m-3)^2
Step-by-step explanation:
(m^2-6m+9) is factored by using this formula:
(a-b)^2=a^2-2ab+b^2.
Therefore, we can substitute the values into this equation.
m^2=a^2,
-6m=-2ab,
9=b^2.
Solving for b in the third equation, b is equal to 3 or -3. However, as this equation is negative, then it must be -3.
(m-3)^2
answer is c please help its about limit
Using limits the value of k is c √2
What is the limit of a function?The limit of a function is the valuer the function tends to as the dependent variable tends to a particular value.
Given that the limit lim n → ∞ [x√(x + 1)[1 - √(2x + 3)]/(7 - 6x + kx²) = - 1, we desirte to find the value of k.
So, we proceed as follows
lim n → ∞ [x√(x + 1)[1 - √(2x + 3)]/(7 - 6x + kx²) = - 1
Factorizing out √2x, we have that
lim n → ∞ [x√(x + 1)[1 - √(2x√(1 + 3/√(2x)]/(7 - 6x + kx²) = - 1
lim n → ∞ [x√(x + 1)√(2x[1/√(2x - √(1 + 3/√(2x)]/(7 - 6x + kx²) = - 1
Also, factorizing out √x, we have that
lim n → ∞ [x√x√(1 + 1/√x)√2x[1/√(2x - √(1 + 3/√(2x)]/(7 - 6x + kx²) = - 1
lim n → ∞ [x√x√2x√(1 + 1/√x)[1/√(2x - √(1 + 3/√(2x)]/(7 - 6x + kx²) = - 1
lim n → ∞ [x√2x√(1 + 1/√x)[1/√(2x - √(1 + 3/√(2x)]/(7 - 6x + kx²) = - 1
lim n → ∞ [√2x²√(1 + 1/√x)[1/√(2x - √(1 + 3/√(2x)]/(7 - 6x + kx²) = - 1
Factorizing x² from the denominator, we have that
lim n → ∞ [√2x²√(1 + 1/√x)[1/√(2x - √(1 + 3/√(2x)]/x²(7/x² - 6/x + k) = - 1
lim n → ∞ [√2√(1 + 1/√x)[1/√(2x - √(1 + 3/√(2x)]/(7/x² - 6/x + k) = - 1
Now substituting x = ∞ into the equation, we have that
[√2√(1 + 1/√∞)[1/√(2∞ - √(1 + 3/√(2∞)]/(7/∞² - 6/∞ + k) = - 1
[√2√(1 + 0)[0 - √(1 + 0]/(0 - 0 + k) = - 1
[√2√(1)[- √1]/k = - 1
[√2(1)[- 1]/k = - 1
-√2/k = - 1
k = -√2/-1
k = √2
So, the value of k is c √2
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