Let's begin by listing out the information given to us:
[tex]\begin{gathered} f\mleft(x\mright)=\frac{1}{5}x+6 \\ f(x)=y \\ y=\frac{1}{5}x+6 \end{gathered}[/tex]To find the inverse, of this equation, we have to interchange the two variables (x for y). We have:
[tex]\begin{gathered} y=\frac{1}{5}x+6\Rightarrow x=\frac{1}{5}y+6 \\ x=\frac{1}{5}y+6 \\ \text{Multiply through each element by 5, we have:} \\ 5\cdot x=\frac{1}{5}y\cdot5+6\cdot5 \\ 5x=y+30 \\ \text{Subtract 30 from both side, we have:} \\ 5x-30=y+30-30 \\ 5x-30=y\Rightarrow y=5x-30 \\ y=5x-30 \\ f(^{-1})=5x-30 \end{gathered}[/tex]Solve. Write in scientific notation. 1.44 x 108 1.2 x 105
1.44 x 1081.2 x 105 =
A penny-farthing is a bicycle with a very large front wheel and a much smaller back wheel. Penny-farthings were popular in the 1800s and were available in different sizes. The ratio of the diameter of the front wheel of a penny-farthing to the diameter of the back wheel is 13:4. What is the ratio of the circumference of the front wheel to the circumference of the back wheel? Explain.
The Solution:
It is given in the question that the ratio of the diameter of the front wheel of a penny-farthing to the diameter of the back wheel is 13:4
[tex]\begin{gathered} \frac{D}{d}=\frac{13}{4} \\ \text{Where} \\ D=\text{diameter of the front wheel} \\ d=\text{diameter of the back wheel} \end{gathered}[/tex]We are required to find the ratio of the circumference of the front wheel to the circumference of the back wheel.
Step 1:
The formula for the circumference of a wheel (that is, a circle) is
[tex]\text{ circumference of a wheel = 2}\pi r=\pi d[/tex]Step 2:
We shall find the ratio of the circumference of the front wheel to the circumference of the back wheel.
[tex]\begin{gathered} \frac{\pi D}{\pi d}=\frac{13\pi}{4\pi}=\frac{13}{4} \\ \text{ So,} \\ 13\colon4 \end{gathered}[/tex]Therefore, the required ratio is 13:4
Simplify using expressions 15m^3n^5 / 3m^2n^2
Simplify using expressions
15m^3n^5 / 3m^2n^2
1. The number
15/3 = 5
____________________
2. m
m^3 / m^2 = m*m*m/m*m = m
____________________
3. n
n^5 /n^2 = n^3
_______________
The simplified expression is 5mn^3
________________________
n^a /n^b = n^(a-b)
n^a x n^b = n^(a+b)
what is the value of the exponents of x in the simplify expression?
Let's use the following property:
[tex]x^y\cdot x^z=x^{y+z}[/tex][tex](x^{-3}y^5z^{-4})\cdot(x^6y^{-7}z^{-2})=x^{-3+6}y^{5-7}z^{-4-2}=x^3y^{-2}z^{-6}[/tex]The school population for a certain school is predicted to increase by 50 students per year for the next 14 years. If the current enrollment is 600 students, what will the enrollment be after 14 years?
Suppose that the school population is 600 students, and that it is predicted to increase by 50 students per year for the next 14 years. Thus we will have:
[tex]\begin{gathered} \text{Year 0: }600 \\ \text{Year 1: }600+50=600+1\cdot50 \\ \text{Year 2: }600+50+50=600+2\cdot50 \\ \ldots \\ \text{Year 14: }600+\underbrace{50+\cdots+50}=600+50\cdot14=600+700=1300 \end{gathered}[/tex]This means that the enrollment after 14 years will be of 1300 students in total.
an entry commercial break 3.6 minutes if each commercial takes 0.6 minutes ,how many commercials will beplayed
We have
an entry commercial break 3.6 minutes
each commercial takes 0.6 minutes
Then we need to divide 3.6 between 0.6 in order to know how many commercials
[tex]\frac{3.6}{0.6}=6\text{ }[/tex]6 commercials in 3.6 minutes
g(t)=t^2 - 2f(t) = 4t+4Find g(t)/f(t)
how many terms are there in each of the following sequences?:
a) The given sequence is expressed as
52, 53, 54, 55, .......252
The first step is to determine the type of sequence by comparing the consecutive terms. We can see that there is a common difference, d between the consecutive terms.
d = 53 - 52 = 54 - 53 = 1
This means that it is an arithmetic sequence. The formula for determining the nth term of an arithmetic sequence is expressed as
an = a1 + (n - 1)d
where
an is the nth term of the sequence
n is the number of terms in the sequence
d is the common difference
a1 is the first term
From the information given,
a1 = 52, d = 1, an = 252
thus, we have
252 = 52 + (n - 1)1
252 = 52 + n - 1
252 = 52 - 1 + n = 51 + n
n = 252 - 51
n = 201
There are 201 terms in the sequence
we have to write the equation of the line using y=mx+b passing through the points (6,2) and (2,4)
The general equation of line passes through point (x_1,y_1) and (x_2,y_2) is,
[tex]y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)[/tex]Determine the equation of line passes through point (6,2) and (2,4).
[tex]\begin{gathered} y-2=\frac{4-2}{2-6}(x-6) \\ y-2=\frac{2}{-4}(x-6) \\ y=-\frac{1}{2}x+3+2 \\ =-\frac{1}{2}x+5 \end{gathered}[/tex]So equation of line is y = -1/2x + 5.
Translate the sentence into an inequality.Twice the difference of a number and 2 is at least - 29.Use the variable y for the unknown number.
We will start translating the phrase "the difference of a number and 2", using y as the unknown number:
[tex]y-2[/tex]Next, using that expression, we translate the phrase "twice the difference of a number and 2", in this step, we multiply the whole previous expression by 2:
[tex]2(y-2)[/tex]To continue we consider the phrase "is at least -29", this means that the previous expression 2(y-2) has to be at least -29 or it can be greater than -29. This is represented in the following expression:
[tex]2(y-2)\ge-29[/tex]Where the symbol ≥ means greater or equal to.
Answer:
[tex]2(y-2)\ge-29[/tex]A carpenter is building a set of trusses to support the roof of a residential home. In theblueprints, she has determined that she needs to make a support triangle with an area 56 m². She knows that the base must be 1 less than 2 times the height. Write the equation thatcorrectly shows the area of the triangle in terms of its height, h.
We are told that we want a triangle of area 56. Recall that the area of a triangle of base b and height h is given by the formula
[tex]\frac{b\cdot h}{2}[/tex]In our case we want
[tex]\frac{b\cdot h}{2}=56[/tex]now, we want to find an expression for b. We are told that the base is one less than twice the height. That is, we take the height, multiply it by 2, and then subtract 1. That would lead to
[tex]b=2h\text{ -1}[/tex]so we have
[tex]\frac{h(2h\text{ -1\rparen}}{2}=56[/tex]so the second option is correct.
11. Which set of points could you use to create a line with slope of -3/2? (A) (5,7) (7,4) (B) (-1,4) (1,7) (C) (3,2) (1,-3)(D) (-3,0) (0,-2)
Answer:
(A) (5,7) (7,4)
Explanation:
To determine the set of points that could be used to create a line with a slope of -3/2, we use the slope formula below.
[tex]\text{Slope}=\frac{Change\text{ in y-axis}}{Change\text{ in x-axis}}[/tex]Option A
[tex]\begin{gathered} \text{Slope}=\frac{7-4}{5-7} \\ =-\frac{3}{2} \end{gathered}[/tex]Therefore, the set of points is (5,7) and (7,4).
find volume of the right triangular prism round to hundred
V = 594mm^3
In order to calculate the volume of a prisma you must multiply the area of the base times the height. The base is a triangle.
However, we don't have the value of the height in it, but we find it by solving the Pythagorean theorem on the triangle.
Match an appropriate graph to each equation. t (x) = 1/x+3t (x)= -1/x+3
t (x) = 1/x + 3
t (x) ⇒ y
y = 1/x + 3
when x = 1
y = 1/1 + 3 = 1 + 3 = 4
y = 4(Graph 4)
t (x)= -1/x + 3
t (x) ⇒ y
y = -1/x + 3
when x = 1
y =-1/1 + 3 = - 1 + 3 = 2
y = 2(Graph 2 )
Hence, the correct graph to these equations is Graph 4 & Graph 2
if x varies directly as y, and x = 10 when y = 5, find x when y = 9. x = ______
Solution:
Since x varies directly as y, consider the following diagram:
by cross-multiplication, we get:
[tex]5x\text{ = (9)(10)}[/tex]this is equivalent to:
[tex]5x\text{ = 90}[/tex]solving for x, we get:
[tex]x\text{ = }\frac{90}{5}=18[/tex]so that, we can conclude that the correct answer is:
x = 18.
Which equation represents the line that is perpendicular to y = 3/4x+ 1 and passes through (-5,11)A.y=-4/3x+13/3B.y=-4/3x+29/3C.y=3/4x+59/4D.y=3/4x-53/4
Answer:
A.y=-4/3x+13/3
Step-by-step explanation:
The equation of a line has the following format:
y = ax + b
In which a is the slope.
Perpendicular to y = 3/4x+ 1:
Two lines are perpendicular if the multiplication of their slopes is -1.
Here the slope is 3/4.
In the answer to this exercise, the slope is a.
So
[tex]\frac{3}{4}\ast a=-1[/tex][tex]\frac{3a}{4}=-1[/tex]Now, cross multiplication
3a = -4
a = -4/3
So, for now, the equation is:
y = (-4/3)x + b
Passes through (-5,11):
This means that when x = -5, y = 11. So
11 = (-4/3)*(-5) + b
11 = (20/3) + b
b = 11 - (20/3)
[tex]11-\frac{20}{3}=\frac{\frac{3}{1}\ast11-\frac{3}{3}\ast20}{3}=\frac{33-20}{3}=\frac{13}{3}[/tex]So the correct answer is:
A.y=-4/3x+13/3
A bus travel's at an average speed of 65.1 miles in 3 hours in the city. how far could the bus travel in 8.2 hours?
177.94miles
Explanations:From the question, distance is directly proportional to the time taken. Mathematically;
[tex]\begin{gathered} d\alpha t \\ d=kt \\ k=\frac{d}{t} \end{gathered}[/tex]where:
d is the distance traveled
t is the time
If the bus travel's at an average speed of 65.1 miles in 3 hours in the city, the variation constant "k" is calculated as;
[tex]\begin{gathered} k=\frac{65.1\text{miles}}{3\text{hours}} \\ k=\frac{21.7mi}{hr} \end{gathered}[/tex]In order to determine how far could the bus travel in 8.2 hours
[tex]\begin{gathered} d=kt \\ d=\frac{21.7miles}{\cancel{hr}}\times8.2\cancel{\text{hrs}} \\ d=177.94\text{miles} \end{gathered}[/tex]Therefore the bus can travel for 177.94miles in 8.2hours
In the figure below, AB AD andAC bisects A. Solve for x. Then, using that value, find the length of AC
Since AB = AD
15x + 4 = 2x + 160
15x - 2x =160 - 4
13x = 156
x =156/13
x = 12
AC = 11X + 35
Since x = 12
AC = 11 x 12 + 35
AC = 132 + 35
AC =167
Select the correct answer. A pyramid is placed inside a rectangular prism with height h. Area of the base of the pyramid and the prism is B. A pyramid is placed inside a prism as shown. The pyramid has the same base area, B, as the prism but half the height, h, of the prism. Which expression gives the volume of the pyramid? A. V = 2 3 B h B. V = 1 4 B h C. V = 2 B h D.
An expression which gives the volume of this pyramid is D. V = 1/6 Bh.
How to calculate the volume of a pyramid?Mathematically, the volume of a pyramid can be calculated by using this formula:
V = 1/3 × B × h
Where:
V represents the volume of a pyramid.h represents the height of a pyramid.B represents the base area of a pyramid.Since the pyramid has the same base area (B) as the prism but half (1/2) the height (h) of the prism, the new volume of this pyramid can be derived as follows:
New height, h = h/2
Substituting the parameters into the formula, we have:
Volume, V = 1/3 × B × (h/2)
Volume, V = 1/6 Bh
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Complete Question:
A pyramid is placed inside a prism as shown. The pyramid has the same base area, B, as the prism but half the height, h, of the prism. Which expression gives the volume of the pyramid?
A. V = 2/3 Bh
B. V = 1/4 Bh
C. V = 2 Bh
D. V = 1/6 Bh
hello can you help me i think the answer is 2
From the graph given,
The Swans are on the x-axis and the Geese are on the y-axis
Where there are 5 swans, the number of geese are 2
Hence, there are 2 geese when there are 5 swans.
Answer:
2 geese
Step-by-step explanation:
Reading a graph
The input is swans and the output is gees
When there is 5 on the x axis, there is 2 on the y axis
For 5 swans, there are 2 geese
if Df and Gi are parallel lines and m≤ihj=60° what is m≤FEH
Answer:
m∠FEH = 60
Explanation:
Angle IHJ and angle FEH are corresponding angles, they are in the same relative position to the parallel lines and the diagonal.
Corresponding angles have the same measure, so the measure of angle FEH is:
m∠FEH = m∠IHJ
m∠FEH = 60
I need help with this. You could select more than one answer
given expression is,
[tex]30x^2-5x-10[/tex]to find the expression equal to the given expression.
the expression is,
[tex]\begin{gathered} -5(-6x^2+x+2) \\ =-5(-6x^2)-5x-5\cdot \\ =30x^2-5x-10 \end{gathered}[/tex]Question 3 of 8
What is the length of CD?
B
15- X
с хр
5
E
20
1
Answer here
Explanation
Triangles ABC and CDE are congruent, then:
[tex]\begin{gathered} \frac{15-x}{20}=\frac{x}{5} \\ 5(15-x)=20x \\ 75-5x=20x \\ 75=20x+5x \\ 75=25x \\ \frac{75}{25}=x \\ 3=x \end{gathered}[/tex]Answer
x=3
Find the X intercept and coordinate of the vertex for the parabola Y=X^2+ 4X -21 ,if there is more than one Y intercept separate them with commas.
In this case, we'll have to carry out several steps to find the solution.
Step 01:
Data:
y = x² + 4x - 21
Step 02:
parabola equation:
y = x² + 4x - 21
a = 1
b = 4
c = -21
x-intercepts:
x² + 4x - 21 = 0
(x + 7)(x - 3) = 0
x1 = - 7
x2 = 3
(-7 , 0)
(3 , 0)
vertex:
[tex]xv\text{ = }\frac{-b}{2a}=\frac{-4}{2\cdot1}=-2[/tex][tex]\begin{gathered} yv=xv^2+4xv\text{ -21} \\ yv=(-2)^2+4(-2)\text{ - 21 = - 25} \end{gathered}[/tex](xv , yv)
(- 2, -25)
The answer is:
x-intercepts:
(-7 , 0)
(3 , 0)
vertex:
(- 2, -25)
Which function is undefined for x = 0?O y=³√x-2Oy=√x-2O y=³√x+2Oy=√x+2
The above function is defined for (x=0)
From the question, we have
Function 1 - y = ∛x-2
Function 2 - y = √x-2
Function 3 - y = ∛x+2
Function 4 - y = √x+2
substituting (x = 0) to determine which function is undefined for (x = 0).
Function 1 - y = ∛x-2
substituting (x = 0), we get
y=∛-2
The above function is defined for (x=0).
Function 2 - y = √x-2
substituting (x = 0), we get
y = √-2
The above function is defined for (x=0).
Function 3 - y = ∛x+2
substituting (x = 0), we get
y = ∛2
The above function is defined for (x=0).
Function 4 - y = √x+2
substituting (x = 0), we get
y = √2
The above function is defined for (x=0).
Hence, it can be concluded that the above function is defined for (x=0)
Subtraction:
Subtraction represents the operation of removing objects from a collection. The minus sign signifies subtraction −. For example, there are nine oranges arranged as a stack (as shown in the above figure), out of which four oranges are transferred to a basket, then there will be 9 – 4 oranges left in the stack, i.e. five oranges. Therefore, the difference between 9 and 4 is 5, i.e., 9 − 4 = 5. Subtraction is not only applied to natural numbers but also can be incorporated for different types of numbers.
The letter "-" stands for subtraction. Minuend, subtrahend, and difference are the three numerical components that make up the subtraction operation. A minuend is the first number in a subtraction process and is the number from which we subtract another integer in a subtraction phrase.
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VFind the area of the figure. (Sides meet at right angles.)3 in5 in5 in10 in8 in
Given the shown composite figure
We will find the area of the figure using the following figure
as shown, the figure is divided into 2 shapes
shape (1) is a rectangle with dimensions 3 in and 5 in
The area of shape (1) = 3 x 5 = 15 in²
shape (2) is a rectangle with dimensions 8 in and 5 in
The area of shape (2) = 8 x 5 = 40 in²
The total area of the figure = 15 + 40 = 55 in²
So, the answer will be Area = 55 in²
How many pounds of candy that sells for $0.87 per lb must be mixed with candy that sells for $1.22 per lb to obtain 9 lb of a mixture that should sell for $0.91 per lb?$0.87-per-lb candy: _____lb$1.22-per-lb candy: _____lb(Type an integer or decimal rounded to two decimal places as needed.)
Let x and y be the candy pounds that sells for $0.87 and $1.22 , respectively. Since they both must add up to 9 lb, we have
[tex]x+y=9...(A)[/tex]On the other hand, the mixture should sell for $0.91 per lib, so we can write,
[tex]0.87x+1.22y=9\times0.91[/tex]Or euivalently,
[tex]\begin{gathered} \frac{0.87}{0.91}x+\frac{1.22}{0.91}y=9 \\ that\text{ is, } \\ 0.95604x+1.340659y=9...(B) \end{gathered}[/tex]Then, we need to solve the following system of equations:
[tex]\begin{gathered} x+y=9...(A) \\ 0.95604x+1.340659y=9 \end{gathered}[/tex]Solving by elimination method.
In order to eliminate variable x, we can to multiply equation (A) by -0.95604 and get an equivalent system of equations:
[tex]\begin{gathered} -0.95604x-0.95604y=-8.60439 \\ 0.95604x+1.340659y=9 \end{gathered}[/tex]Then, by adding both equations, we get
[tex]0.384619y=0.39561[/tex]Then, y is given by
[tex]\begin{gathered} y=\frac{0.39561}{0.384619} \\ y=1.02857 \end{gathered}[/tex]Once we have obtained the result for y, we can substitute in into equation (A), that is,
[tex]x+1.02857=9[/tex]then, x is given as
[tex]\begin{gathered} x=9-1.02857 \\ x=7.9714 \end{gathered}[/tex]Therefore, by rounding to two decimal places, the answer is:
$ 0.87 per lb of candy: 7.97 lb
$1.22-per-lb of candy: 1.03 lb
7. (-/5 Points]DETAILSMY NOTESASMaurice is traveling to Mexico and needs to exchange $390 into Mexican pesos. If each dollar is worth 12.29 pesos, how many pesos will he get for his trip?pesos
Let's apply Rule of 3
1 dollar ---------------- 12.29 Mexican pesos
390 dollars----------- x
x= 390 . 12.29 = 4793.1 Mexican pesos
Instructions: Use the given information to answer the questions and interpret key features. Use any method of graphing or solving. Round to one decimal place, if necessary.
SOLUTION
The given equation is:
[tex]h(x)=-x^2+10x+9.5[/tex]The graph of the function is shown.
The irrigation system is positioned 9.5 feet above the ground to start
The spray reaches maximum height of 34.5 feet at a horizontal distance of 5 feet away from the sprinkler head.
The spray reaches all the way to the ground about 10.874 feet away.
Gerardo is skiing on a circular ski trail that has a radius of 0.9 km. Gerardo starts at the 3-o'clock position and travels 2.6 km in the counter-clockwise direction.How many radians does Gerardo sweep out? ______radians When Gerardo stops skiing, how many km is Gerardo to the right of the center of the ski trail?______ km When Gerardo stops skiing, how many km is Gerardo above of the center of the ski trail? ____km
Circle and Angles
Gerardo is skiing on a trail that has a radius of r = 0.9 km
He starts skiing at the 3-o'clock position. This means he is initially at the right of the center of the circular trail. This position corresponds to the zero degrees (or radians) reference.
The arc length of a circle of radius r is given by:
[tex]L=\theta r[/tex]Where θ is the central angle in radians.
We know Gerardo travels L=2.6 km in the counter-clockwise direction, thus the angle is calculated by solving for θ:
[tex]\theta=\frac{L}{r}\text{ }[/tex]Substituting:
[tex]\theta=\frac{2.6}{0.9}=2.8889rad\text{ }[/tex]Gerardo swept out 2.8889 radians.
Now we need to calculate the rectangular coordinates of the final position where Gerardo stopped skiing. Since the angle is less than one turn of the trail, and the angle is measured counter-clockwise, we can use the formulas:
x = r cos θ
y = r sin θ
Substituting:
x = 0.9 cos 2.8889 rad
x = -0.87 km
y = 0.9 sin 2.8889 rad
y = 0.23 km
Gerardo is -0.87 km to the right of the center. In fact, he is 0.87 km to the left of the center.
Gerardo is 0.23 km above the center of the ski trail.