Answer:
2 outcomes
Step-by-step explanation:
Let's list count all the possible outcomes:
(1,3) (1,4) (1,5) (1,6)
(2,3) (2,4) (2,5) (2,6)
(3,3) (3,4) (3,5) (3,6)
As expected, there are 12 (3x4) possible outcomes.
How many outcomes do not show an even number (so showing 1 or 3) on the first spinner and show a 6 on the second spinner?
There are two cases where 6 is on the second spinner and NOT an even number on the first spinner: (1,6) and (3,6)
translate this sentence into an inequality. A cheetah can reach a speed of 70 mph. however, this speed can be maintained for no more than 1,640 feet. A. d>1,640 B. d_<1,640 C. d<1,640 D. d_>1,640
The answer is:
B. [tex]d\leq 1,640ft[/tex]
Why?From the statement we know that the cheetah can reach a speed of 70 mph, but it can be maintained for no more than 1,640 feet.
The expression "no more than" means that at least it can be reached but never exceeded, it involves that the distance can be less or equal than 1,640 feet but never more than that.
So, the correct option is:
B. [tex]d\leq 1,640ft[/tex]
Have a nice day!
Use the three steps to solve the problem.
The sum of 3 consecutive Integral numbers is 117. Find the numbers.
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Answer:
The numbers are 38, 39, and 40
Step-by-step explanation:
Let the 3 consecutive Integral numbers be;
x, x+1, x+2
We are informed that the sum of the 3 consecutive Integral numbers is 117. Therefore;
x + x+1 + x + 2 = 117
3x + 3 = 117
3x = 114
x = 38
x+1 = 39
x+2 = 40
Answer:
38, 39, 40
Step-by-step explanation:
Here's an interesting solution to this one. The formula to compute the nth triangular number, which is to say the sum of the first n consecutive integers, starting at 1 is
[tex]\frac{n(n+1)}{2}[/tex]
What if we wanted to start from a higher number, though? Say, 3. Well, we'd have to shift every number in the sequence up 2 (1, 2, 3 would become 3, 4, 5) so we'd be adding 2 n times. If we wanted to be more general, we could call that "shift amount" s, and our modified formula would now look like
[tex]\frac{n(n+1)}{2}+sn[/tex]
Now let's put this formula to the test. We know what our sum is here: it's 117. And we know what our n is too; we're finding 3 integers, so n = 3. This gives us the equation
[tex]\frac{3(3+1)}{2} +3s=117[/tex]
Solving this equation for s:
[tex]\frac{3(4)}{2} +3s=117\\\\\frac{12}{2}+3s=117\\ 6+3s=117\\3s=111\\s=37[/tex]
so our "shift amount" is 37, and our sequence gets shifted from 1, 2, 3 to 38, 39, 40.
But why?This was a lot of setup for what seems like a disappointing payoff, but the real power with this approach is that we've actually just solved every problem of this type. Let's say you had to find the sum of 5 consecutive integers, and their sum was 70. Not a problem. Just set our n = 5 and solve:
[tex]\frac{5(6)}{2} +5s=70\\\\\frac{30}{2} +5s = 70\\15+5s=70\\5s=55\\s=11[/tex]
Which gives us a "shift" of 11 and the sequence 12, 13, 14, 15, 16 (which is exactly the sequence I came up with for this problem!)
The area of a rhombus is 40 in. If
one diagonal of the rhombus is 8 in,
what is the length of the other
diagonal?
Answer:
10 in.
Step-by-step explanation:
The area of a rhombus is the product of the lengths of the diagonals divided by 2.
Let the diagonals be x and y.
area = xy/2
Here you have
area = 40 in.^2
x = 8 in.
We are looking for y, the other diagonal.
xy/2 = area
(8 in.)y/2 = 40 in.^2
(8 in.)y = 80 in.^2
y = 10 in.
Answer: The other diagonal has length 10 in.
Answer is provided in the image attached.
Mack plans to meet his 4 friends. How many different ways can he make his visit if he visits each friends once?
Answer:
24
Step-by-step explanation:
Mack can choose any of the 4 for the first visit, any of the remaining 3 for the second visit, either of the remaining 2 for the third visit, then visit the last one. There are 4·3·2·1 = 24 ways Mack can do this.
_____
The number 4·3·2·1 is "four factorial", written as 4! (with an exclamation point). It is the number of ways 4 objects can be ordered, called the number of permutations of 4 objects.
would be nice to if somebody helped me.
Answer:
1,456
Step-by-step explanation:
The sum of n terms of a geometric sequence with first term a1 and common ratio r is given by ...
Sn = a1·(r^n -1)/(r -1)
For your series with a1=4, r=3, and n=6, the sum is ...
S6 = 4·(3^6 -1)/(3 -1) = 2·728 = 1,456
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Answer:
a)
[tex]y=400(2.5)^{x}[/tex]
b)
3,814,698
c)
16.08 weeks
Step-by-step explanation:
a)
The question presented here is similar to a compound interest problem. We are informed that there are 400 rice weevils at the beginning of the study. In a compound interest problem this value would be our Principal.
P = 400
Moreover, the population is expected to grow at a rate of 150% every week. This is equivalent to a rate of interest in a compound interest problem.
r = 150% = 1.5
The compound interest formula is given as;
[tex]A=P(1+r)^{n}[/tex]
We let y be the weevil population in any given week x. The formula that can be used to predict the weevil population is thus;
[tex]y=400(1+1.5)^{x}\\\\y=400(2.5)^{x}[/tex]
b)
The weevil population 10 weeks after the beginning of the study is simply the value of y when x = 10. We substitute x with 10 in the equation obtained from a) above;
[tex]y=400(2.5)^{10}\\\\y=3814697.3[/tex]
Therefore, the weevil population 10 weeks after the beginning of the study is approximately 3,814,698
c)
We are simply required to determine the value of x when y is
1,000,000,000
Substitute y with 1,000,000,000 in the equation obtained in a) above and solve for x;
[tex]1000000000=400(2.5)^{x}\\\\2.5^{x}=2500000\\\\xln(2.5)=ln(2500000)\\\\x=\frac{ln(2500000}{ln(2.5)}=16.0776[/tex]
The area of a triangle is 17.5 square meters. The height of the triangle is 3 meters less than twice its base. The base of the triangle is x meters. Complete the equation that represents this description and fill in the values for the base and height of the triangle.
Answer:
17.5 = (1/2)(x)(2x-3)base: 5 m; height: 7 mStep-by-step explanation:
The base is defined as x. The height is said to be 3 less than 2x, so is (2x-3).
The formula for the area of a triangle is ...
A = (1/2)bh
Filling in the given values, we have ...
17.5 = (1/2)x(2x-3)
35 = 2x^2 -3x . . . . multiply by 2
2x^2 -3x -35 = 0 . . . . put in standard form
(2x +7)(x -5) = 0 . . . . . factor
The base is 5 meters; the height is 2·5-3 = 7 meters.
The base and height of the triangle satisfying the given conditions are approximately 4.3 and 5.6 meters, respectively.
Explanation:The area of a triangle is given by the formula 1/2 * base * height. Here, the area is 17.5 square meters, the base of the triangle is x, and the height of the triangle is 3 meters less than twice its base, therefore the height is 2x-3. Plugging these values into the formula, we get 17.5 = 1/2 * x * (2x - 3).
To solve this equation for x, first simplify the right-hand side, yielding 17.5 = x*(2x - 3). Multiplying this out gives 17.5 = 2x^2 - 3x. Then, rearrange to get the equation in standard quadratic form, resulting in 2x^2 - 3x - 17.5 = 0.
Through using quadratic formula we can find the solution(s) to be approximately x = 4.3 or x = -2.0. Since a negative value for x ? the base of a triangle ? is not possible, we discard that solution. Thus, the base of the triangle is 4.3 meters, and the height would then be 2*4.3 - 3 = about 5.6 meters.
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The function f(x)= x - 6x + 9 is shifted 5 units to the left to create g(x). What is
Answer:
g(x) = x^2 + 4x + 4
Step-by-step explanation:
In translation of functions, adding a constant to the domain values (x) of a function will move the graph to the left, while subtracting from the input of the function will move the graph to the right.
Given the function;
f(x) = x2 - 6x + 9
a shift 5 units to the left implies that we shall be adding the constant 5 to the x values of the function;
g(x) = f(x+5)
g(x) = (x+5)^2 - 6(x+5) + 9
g(x) = x^2 + 10x + 25 - 6x -30 + 9
g(x) = x^2 + 4x + 4
(-6,1) is a point on the graph of y=g(x)
What point is on the graph of y=g(x+1)-5?
What point is on the graph of y= -2g(x-2)+4?
What point is on the graph of y=g(2x+2)?
Answer in an ordered pair
Answer:
[tex](-7, -4)[/tex]
[tex](-4, 2)[/tex]
[tex](-4, 1)[/tex]
Step-by-step explanation:
We know that the point (-6, 1) belongs to the main function g(x)
The transformation
[tex]y = g (x + 1) -5[/tex]
add 1 to the input variable (x) and subtract 5 to the output variable (y)
So the point in the graph of [tex]y = g (x + 1) -5[/tex] is
[tex]x + 1 =-6\\x = -7[/tex]
[tex]y= 1-5\\y = -4[/tex]
The point is: [tex](-7, -4)[/tex]
The transformation
[tex]y = -2g(x -2) +4[/tex]
subtract two units from the input variable (x), multiply the output variable (y) by -2 and then add 4 units
So the point in the graph of [tex]y = -2g(x -2) +4[/tex] is
[tex]x -2 =-6\\x = -4\\\\y = -2(1)+4\\y = 2[/tex]
The point is: [tex](-4, 2)[/tex]
The transformation
[tex]y=g(2x+2)[/tex]
Multiply the input variable (x) by 2 and then add two units
So the point in the graph of [tex]y=g(2x+2)[/tex] is
[tex]2x +2 =-6\\2x = -8\\x=-4[/tex]
[tex]y=1[/tex]
The point is: [tex](-4, 1)[/tex]
Final answer:
The transformation y=g(x+1)-5 results in the point (-7, -4), the transformation y=-2g(x-2)+4 gives the point (-4, 2), and the transformation y=g(2x+2) results in (-4, 1) on their respective graphs.
Explanation:
If the point (-6, 1) is on the graph of y=g(x), then we need to find the corresponding points for the given transformations of the function g(x).
For the function y=g(x+1)-5, the x-coordinate will be shifted left by 1, and the y-coordinate will be 5 less than the original y value. Therefore, the new point will be (-7, -4).
In the case of y=-2g(x-2)+4, the x-coordinate will be shifted right by 2, and the y value will be scaled by a factor of -2 and increased by 4. If g(x) was 1 when x was -6, then for x-2, g(x) would be 1 when x is -4. So, we plug in the original x value of -6 into this transformation to get (-4, -2*1+4), which simplifies to (-4, 2).
For y=g(2x+2), we find the new x-coordinate by setting 2x+2 = -6, which gives x = -4. The new point does not change the y-coordinate as there's no vertical shift, so the point is (-4, 1).
These transformations illustrate the dependence of y on x and show how function composition and arithmetic operations alter the input-output pairs in a function's graph.
The point C is rotated 90° clockwise around the origin what are the coordinates of the resulting. Z’
Answer:
Z'(-2, 3)
Step-by-step explanation:
For 90° CW rotation, the transformation is ...
(x, y) ⇒ (y, -x)
Z(-3, -2) ⇒ Z'(-2, 3)
) Set up a double integral for calculating the flux of F=3xi+yj+zk through the part of the surface z=−5x−2y+2 above the triangle in the xy-plane with vertices (0,0), (0,2), and (2,0), oriented upward. Instructions: Please enter the integrand in the first answer box. Depending on the order of integration you choose, enter dx and dy in either order into the second and third answer boxes with only one dx or dy in each box. Then, enter the limits of integration and evaluate the integral to find the flux.
Final answer:
To calculate the flux of the vector field through the given surface, set up a double integral using the dot product of the field and the unit normal vector. Choose the order of integration and determine the limits of integration based on the given triangle in the xy-plane. The double integral will be evaluated to find the flux.
Explanation:
To set up a double integral for calculating the flux of F=3xi+yj+zk through the given surface, we need to determine the limits of integration and the order of integration. Since the triangle in the xy-plane has vertices (0,0), (0,2), and (2,0), the limits of integration for x and y will be from 0 to 2. The order of integration can be either dx dy or dy dx, but let's choose dx dy for this problem.
Therefore, the integrand is the dot product of F and the unit normal vector n to the surface: (3x, y, 1) • (-5, -2, 1). So the integrand is -15x-2y+z.
The limits of integration are x = 0 to x = 2 and y = 0 to y = 2 - x. The double integral to find the flux is:
∫∫R (-15x-2y+z) dx dy, where R represents the region defined by the triangle in the xy-plane.
Match the systems of linear equations with their solutions.
Answer:
The solutions of linear equations in the procedure
Step-by-step explanation:
Part 1) we have
x+y=-1 ----> equation A
-6x+2y=14 ----> equation B
Solve the system by elimination
Multiply the equation A by 6 both sides
6*(x+y)=-1*6
6x+6y=-6 -----> equation C
Adds equation C and equation B
6x+6y=-6
-6x+2y=14
-------------------
6y+2y=-6+14
8y=8
y=1
Find the value of x
substitute in the equation A
x+y=-1 ------> x+1=-1 ------> x=-2
The solution is the point (-2,1)
Part 2) we have
-4x+y=-9 -----> equation A
5x+2y=3 ------> equation B
Solve the system by elimination
Multiply the equation A by -2 both sides
-2*(-4x+y)=-9*(-2)
8x-2y=18 ------> equation C
Adds equation B and equation C
5x+2y=3
8x-2y=18
----------------
5x+8x=3+18
13x=21
x=21/13
Find the value of y
substitute in the equation A
-4x+y=-9 ------> -4(21/13)+y=-9 ----> y=-9+84/13 -----> y=-33/13
The solution is the point (21/13,-33/13)
Part 3) we have
-x+2y=4 ------> equation A
-3x+6y=11 -----> equation B
Multiply the equation A by 3 both sides
3*(-x+2y)=4*3 ------> -3x+6y=12
so
Line A and Line B are parallel lines with different y-intercept
therefore
The system has no solution
Part 4) we have
x-2y=-5 -----> equation A
5x+3y=27 ----> equation B
Solve the system by elimination
Multiply the equation A by -5 both sides
-5*(x-2y)=-5*(-5)
-5x+10y=25 -----> equation C
Adds equation B and equation C
5x+3y=27
-5x+10y=25
-------------------
3y+10y=27+25
13y=52
y=4
Find the value of x
Substitute in the equation A
x-2y=-5 -----> x-2(4)=-5 -----> x=-5+8 ------> x=3
The solution is the point (3,4)
Part 5) we have
6x+3y=-6 ------> equation A
2x+y=-2 ------> equation B
Multiply the equation B by 3 both sides
3*(2x+y)=-2*3
6x+3y=6
so
Line A and Line B is the same line
therefore
The system has infinite solutions
Part 6) we have
-7x+y=1 ------> equation A
14x-7y=28 -----> equation B
Solve the system by elimination
Multiply the equation A by 7 both sides
7*(-7x+y)=1*7
-49x+7y=7 -----> equation C
Adds equation B and equation C
14x-7y=28
-49x+7y=7
------------------
14x-49x=28+7
-35x=35
x=-1
Find the value of y
substitute in the equation A
-7x+y=1 -----> -7(-1)+y=1 ----> y=1-7 ----> y=-6
The solution is the point (-1,-6)
The diagram is not to scale.
Answer:
[tex]|AB|=30[/tex]
Step-by-step explanation:
From the diagram,
AO=OC=16 units, all radii of a circle are equal.
BO=OC+BC
BO=16+18
BO=34
A tangent to a circle will always meet the radius at right angles.
We use the Pythagoras Theorem to obtain:
[tex]|AB|^2+|AO|^2=|BO|^2[/tex]
[tex]|AB|^2+16^2=34^2[/tex]
[tex]|AB|^2+256=1156[/tex]
[tex]|AB|^2=1156-256[/tex]
[tex]|AB|^2=900[/tex]
Take positive square roots to get:
[tex]|AB|=\sqrt{900}[/tex]
[tex]|AB|=30[/tex]
Honolulu covers an area of 68.4 square miles. There are approximately 348,000 people living in Honolulu. Anchorage, Alaska has an area of 1706 square miles and has a population of approximately 298,000 people. How many more people, per square mile, live in Honolulu verses Anchorage? Round to the nearest person per square mile.
Answer:
4,913 people more per sq mile.
Step-by-step explanation:
First step is to calculate the population density of both cities, then we'll be able to answer the question.
We're looking for a number of people per sq mile... so we'll divide the population by the area.
Honolulu: 348,000 people on 68.4 sq miles
DensityH = 348,000 / 68.4 = 5,088 persons/sq mile
Anchorage: 298,000 people on 1,706 sq miles
DensityA = 298,000 /1 ,706 = 175 persons/sq mile
Then we do the difference... 5,088 - 175 = 4,913 more people per sq mile.
Honolulu has approximately 4913 more people per square mile compared to Anchorage.
Explanation:To find the number of more people per square mile in Honolulu compared to Anchorage, we need to calculate the population density of each city. Population density is calculated by dividing the population by the area.
For Honolulu:
Population density = population / area = 348,000 / 68.4 = 5087.72 people per square mile.
For Anchorage:
Population density = population / area = 298,000 / 1706 = 174.46 people per square mile.
To find the difference, we subtract the population density of Anchorage from the population density of Honolulu:
Difference = 5087.72 - 174.46 = 4913.26 people per square mile.
Rounding to the nearest person per square mile, there are approximately 4913 more people per square mile in Honolulu compared to Anchorage.
Write an Explicit formula for a1 = –2, an = an – 1 + 4, n ≥ 2
A. an = –4n – 6
B. an = –2n – 6
C. an = 4n – 6
D. an = 4n + 6
According to the recursive formula,
[tex]a_2=a_1+4[/tex]
[tex]a_3=a_2+4=(a_1+4)+4=a_1+2\cdot4[/tex]
[tex]a_4=a_3+4=a_2+2\cdot4=a_1+3\cdot4[/tex]
and so on, with the general formula
[tex]a_n=a_1+(n-1)\cdot4[/tex]
Then
[tex]a_n=-2+4(n-1)=4n-6[/tex]
and the answer is C.
Answer:
C. an = 4n -6
Step-by-step explanation:
Only one of the offered choices gives a1=-2 for n=1.
___
The recursive formula tells you ...
a2 -a1 = 4
The only choices that increase by 4 when n increases by 1 are choices C and D. Of these, choice D gives a1=4·1+6 = 10 ≠ -2.
Choice C gives a1 = 4·1 -6 = -2, as required.
Use the Divergence Theorem to compute the net outward flux of the vector field F across the boundary of region D. D is the region between the spheres of radius 4 and 5 centered at the origin. F = <9z+4x, x-7y, y+9z>
By the divergence theorem,
[tex]\displaystyle\iint_{\partial D}\vec F\cdot\mathrm d\vec S=\iiint_D(\nabla\cdot\vec F)\,\mathrm dV[/tex]
We have
[tex]\nabla\cdot\vec F(x,y,z)=\dfrac{\partial(9z+4x)}{\partial x}+\dfrac{\partial(x-7y)}{\partial y}+\dfrac{\partial(y+9z)}{\partial z}=6[/tex]
In the integral, convert to spherical coordinates, taking
[tex]x=u\cos v\sin w[/tex]
[tex]y=u\sin v\sin w[/tex]
[tex]z=u\sin w[/tex]
so that
[tex]\mathrm dV=u^2\sin w\,\mathrm du\,\mathrm dv\,\mathrm dw[/tex]
Then the flux is
[tex]\displaystyle6\int_{w=0}^{w=\pi}\int_{v=0}^{v=2\pi}\int_{u=4}^{u=5}u^2\sin w\,\mathrm du\,\mathrm dv\,\mathrm dw=\boxed{488\pi}[/tex]
The net outward flux of the vector field F across the boundary of region D is 488[tex]\pi[/tex] and this can be determined by using the divergence theorem.
Given :
D is the region between the spheres of radius 4 and 5 centered at the origin. F = <9z+4x, x-7y, y+9z>
According to the divergence theorem:
[tex]\rm \int\int_{\delta D} \bar{F}.d\bar{S} = \int\int\int_D(\bigtriangledown.\bar{F} )dV[/tex]
Now, the expression for [tex]\rm \bigtriangledown .\bar{F}[/tex] is given by:
[tex]\rm \bigtriangledown .\bar{F}(x,y,z)=\dfrac{\delta(9z+4x)}{\delta x}+\dfrac{\delta(x-7y)}{\delta y}+\dfrac{\delta(y+9z)}{\delta z}[/tex]
Now, the spherical coordinates is given by:
x = u cosv sinw
y = u sinv sinw
z = u sinw
Therefore, the value of dV is given by:
[tex]\rm dV = u^2sinw\;du\;dv\;dw[/tex]
Now, the net outward flux of the vector field F across the boundary of region D is given by:
[tex]\rm \int\int_{\delta D} \bar{F}.d\bar{S} =\rm \int^{\pi}_0\int^{2\pi}_0\int^5_4 u^2sinw\;du\;dv\;dw[/tex]
Simplify the above integral.
[tex]\rm \int\int_{\delta D} \bar{F}.d\bar{S} =488\pi[/tex]
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The diameter of a sphere is 12
inches. What is the appropriate
surface are, in square inches, of the
sphere if Surface Area = 4tr2?
Answer:
first u should find the radius .radius is half of diameter 12/2=6 so surface area of sphere is 4*3.142*6*6=452.448 square in
The sum of the numbers x, y, and z is 50. The ratio of x to y is 1:4, and the ratio of y and z is 4:5. What is the value of y?
Answer:
y = 20
Step-by-step explanation:
x+y+z = 50
x/y = 1/4 so y = 4x
Substitute y = 4x into x+y+z = 50
x + 4x + z = 50
5x + z = 50
y/z = 4/5 --> 4z = 5y so z =5/4 y
Substitute z =5/4 y into 5x + z = 50
5x + 5/4 y = 50
You can solve for x from these 2 equations
5x + 5/4 y = 50
y = 4x
Substitute y = 4x into 5x + 5/4 y = 50
5x + 5/4 (4x) = 50
5x + 5x = 50
10x = 50
x = 5
y = 4x = 4 (5) = 20
Answer
y = 20
The sum of the numbers x, y, and z is 50. The ratio of x to y is 1:4, and the ratio of y and z is 4:5
The value of y =20
Given :
The sum of the numbers x, y, and z is 50
The ratio of x to y is 1:4, and the ratio of y and z is 4:5.
The sum of the numbers x, y, and z is 50
The equation becomes [tex]x+y+z=50[/tex]
The ratio of x to y is 1:4
[tex]\frac{x}{y} =\frac{1}{4} \\4x=y\\y=4x[/tex]
Now use the second ratio . the ratio of y and z is 4:5
[tex]\frac{y}{z} =\frac{4}{5}\\5y=4z\\Replace \; y=4x\\5(4x)=4z\\20x=4z\\z=5x[/tex]
Replace y=4x and z=5x in the first equation
[tex]x+y+z=50\\x+4x+5x=50\\10x=50\\x=5[/tex]
Now we replace x with 5 and find out y
[tex]y=4x\\y=4(5)\\\y=20[/tex]
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Which property was use to simplify this expression? (will be marked brainliest)
4 (b+2) = 4b + 8
Distributive property
Commutative property
Associative property
Inverse property
Answer:
Distributive property
Step-by-step explanation:
The distributive property tells you ...
a(b+c) = ab +ac
Here, you have a=4, c=2, so ...
4(b+2) = 4·b + 4·2 = 4b +8
Answer:
Below
Step-by-step explanation:
a(b+c) = ab +ac
4(b+2) = 4·b + 4·2 = 4b +8
Which is distributive property!
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thanks!
Find the inverse of the function below and write it in the formyequals=f Superscript negative 1 Baseline left parenthesis x right parenthesisf−1(x).(b) Verify the relationshipsf(f^-1(x)) and f^-1(f(x))=xf(x)=3x+5(a) f^-1(x)=.....
The inverse of the function f(x) = 3x + 5 is [tex]f^{-1}(x) = \dfrac{x - 5}{3}[/tex]
How to determine the inverse of the function
From the question, we have the following parameters that can be used in our computation:
f(x) = 3x + 5
Express the function as an equation
So, we have
y = 3x + 5
Swap the occurrence of x and y in the equation
This gives
x = 3y + 5
Subtract 5 from all sides
3y = x - 5
So, we have
[tex]y = \dfrac{x - 5}{3}[/tex]
Express as an inverse function
[tex]f^{-1}(x) = \dfrac{x - 5}{3}[/tex]
[tex]f^{-1}(x) = \dfrac{x - 5}{3}[/tex]
Verifying the relationship [tex]f^{-1}(f(x))[/tex] and [tex]f(f^{-1}(x))[/tex]
We have
[tex]f^{-1}(f(x)) = \dfrac{3x + 5 - 5}{3}[/tex]
[tex]f^{-1}(f(x)) = \dfrac{3x }{3}[/tex]
[tex]f^{-1}(f(x)) = x[/tex]
Also, we have
[tex]f(f^{-1}(x)) = 3 * \dfrac{x - 5}{3} + 5[/tex]
[tex]f(f^{-1}(x)) = x - 5 + 5[/tex]
[tex]f(f^{-1}(x)) = x[/tex]
Hence, the inverse of the function is [tex]f^{-1}(x) = \dfrac{x - 5}{3}[/tex]
Question
Find the inverse of the function below and write it in the form y = f^-1(x)
Verify the relationshipsf(f^-1(x)) and f^-1(f(x))=x
f(x)=3x+5
4-2(x+7)=3(x+5) using the equation solver
Answer:
x = -5
Step-by-step explanation:
We don't know what equation solver you're supposed to use. Here are the results from one available on the web.
4 - 2(x + 7) = 3(x + 5)
4 - 2x - 14 = 3x + 15
-2x - 3x = 15 + 14 - 4
-5x = 25
x = 25/(-5)
x = - 5a ball is thrown with a slingshot at a velocity of 110ft/sec at an angle of 20 degrees above the ground from a height of 4.5 ft. approximentaly how long does is take for the ball to hit the ground. Acceleration due to gravity is 32ft/s^2
Answer:
[tex]t=2.47\ s[/tex]
The ball takes 2.47 seconds to touch the ground
Step-by-step explanation:
The equation that models the height of the ball in feet as a function of time is:
[tex]h(t) = h_0 + s_0t -16t ^ 2[/tex]
Where [tex]h_0[/tex] is the initial height, [tex]s_0[/tex] is the initial velocity and t is the time in seconds.
We know that the initial height is:
[tex]h_0 = 4.5\ ft[/tex]
The initial speed is:
[tex]s_0 = 110sin(20\°)\\\\s_0 = 37.62\ ft/s[/tex]
So the equation is:
[tex]h (t) = 4.5 + 37.62t -16t ^ 2[/tex]
The ball hits the ground when when [tex]h(t) = 0[/tex]
So
[tex]4.5 + 37.62t -16t ^ 2 = 0[/tex]
We use the quadratic formula to solve the equation for t
For a quadratic equation of the form
[tex]at^2 +bt + c[/tex]
The quadratic formula is:
[tex]t=\frac{-b\±\sqrt{b^2 -4ac}}{2a}[/tex]
In this case
[tex]a= -16\\\\b=37.62\\\\c=4.5[/tex]
Therefore
[tex]t=\frac{-37.62\±\sqrt{(37.62)^2 -4(-16)(4.5)}}{2(-16)}[/tex]
[tex]t_1=-0.114\ s\\\\t_2=2.47\ s[/tex]
We take the positive solution.
Finally the ball takes 2.47 seconds to touch the ground
It would take approximately B. 2.47 seconds
What is the coefficient in this expression?
5-4.7-2x+5/8
A. -4.7
B. -2
C. 5/8
D. 5
The coefficient is the number with a variable ( letter)
In the given equation you have -2x, where x is the variable, so the coefficient would be -2.
The answer is B.
There are 40 students in an elementary statistics class. On the basis of years of experience, the instructor knows that the time needed to grade a randomly chosen first examination paper is a random variable with an expected value of 6 min and a standard deviation of 6 min.If a grading times are independent and the instructor begins grading at 6:50 P.M. and grades continuously, the (approximate probability that he is through grading before the 11:00 P.M. TV news begins is probability isIf the sports report begins at 11:10 P.M., the probability that he misses part of the report if he waits until grading is done before turning on the TV is
Final answer:
To determine the probability of the instructor finishing grading before 11:00 P.M., calculate the expected time to grade 40 exams, determine the standard deviation, and then use the z-score to find the corresponding probability from the standard normal distribution.
Explanation:
To tackle these statistics problems, there are several concepts we need to apply including expected value, standard deviation, the central limit theorem, hypothesis testing, and probability. Since only one problem can be answered at a time, I'll focus on the first one you've mentioned about the grading times for exams.
The instructor's time to grade each paper is a random variable with an expected value of 6 minutes and a standard deviation of 6 minutes. When considering the grading of 40 papers, we can use the central limit theorem which suggests that the sum of these independent random variables will be approximately normally distributed given the large number of papers (n=40).
We first calculate the expected total time to grade 40 exams by multiplying the individual exam time's expected value by the number of exams: 6 minutes/exam * 40 exams = 240 minutes. Then, we calculate the standard deviation for the total grading time: 6 minutes/exam * √40 ≈ 37.95 minutes.
To find the probability that the instructor finishes grading before 11:00 P.M., we need to calculate the number of minutes from 6:50 P.M. to 11:00 P.M., which is 250 minutes. Next, we convert this problem into a z-score problem where we find the z-score corresponding to 250 minutes. Finally, we look up this z-score in a standard normal distribution table (or use statistical software) to find the corresponding probability.
What is the range of the function y=2e^x-1
all real numbers less than –1
all real numbers greater than –1
all real numbers less than 1
all real numbers greater than 1
Answer:Answer: All real numbers greater than -1. Step-by-step explanation: We have to find the range of: y= 2e^x-1. We know that e^x lies in (0,∞). Hence, 2e^x lies in (0,∞). Hence, 2e^x-1 lies in (-1 ...
Step-by-step explanation:Answer: All real numbers greater than -1. Step-by-step explanation: We have to find the range of: y= 2e^x-1. We know that e^x lies in (0,∞). Hence, 2e^x lies in (0,∞). Hence, 2e^x-1 lies in (-1 ...
Answer:
B. All real numbers greater than –1.
Step-by-step explanation:
We have been given a function [tex]y=2e^x-1[/tex]. We are asked to find the range of our given function.
We know that the range of an exponential function [tex]f(x)=c\cdot n^{ax+b}+k[/tex] is [tex]f(x)>k[/tex].
Upon looking at our given function, we can see that [tex]k=-1[/tex], therefore, the range of our given function would be [tex]y>-1[/tex] that is all real numbers greater than [tex]-1[/tex].
Two similar polygons have areas of 4 square inches and 64 square inches. The ratio of a pair of corresponding sides is 1/4. True False
Answer:
That is true
Step-by-step explanation:
The ratio is a one-to-one measure, literally a ratio of the sides in reduced form. The area is that one-to-one ratio squared.
Our numbers are already squared, so in order to find the one-to-one we have to take the square roots of both of them.
[tex]\frac{\sqrt{4} }{\sqrt{64} } =\frac{2}{8} =\frac{1}{4}[/tex].
Answer:
True
step-by-step explanation:
Just in case you needed a second opinion.
sin F =
The answer ?!?!
Answer:
sin F = a/c
Step-by-step explanation:
The sin ratio by definition for a given angle is the side opposite that angle over the hypotenuse. The only thing that will never change sides in a right triangle is the hypotenuse. In other words, no matter what angle you look at, while the side opposite one angle may be the adjacent side to a different angle, the hypotenuse is always the hypotenuse. That means that c is the hypotenuse in our triangle. The side opposite angle F is a, so the sin of F = a/c.
(I need help as soon as i can! :-) Can you find the third angle measure in a triangle, if you know the other 2 angel measure?
Answer: Yes you can using the sum of internal angles in a triangle they add up to 180, so if you add the two given angle measures and them substract the result from 180 you will have the measure of the third angle. Hope this helps. :)
Step-by-step explanation:
Find the angle between V and W, Round your answer to one decimal place, if necessary V=-5i+2j W=-8i+2j A 17.8° B 353.9° C 3.9° D 7.8°
Answer:
D. 7.8°
Step-by-step explanation:
There are many ways to work this problem. One is to subtract the angle of V from that of W:
∠V = arctan(2/-5) ≈ 158.20°
∠W = arctan(2/-8) ≈ 165.96°
Then ∠W -∠V = 165.96° -158.20° = 7.76° ≈ 7.8°
___
Another is to divide W by V, since the quotient will have an angle that is the difference of their two angles.
(-8i +2j)/(-5i +2j) = (1/29)(44i +6j)
Then the angle of that is ...
arctan(6/44) ≈ 7.8°
___
You can also divide the dot product by the product of the two magnitudes to find the cosine of the angle between the vectors.
(V•W)/(|V|·|W|) = 44/√(68·29) = cos(x)
x = arccos(0.990830168...) ≈ 7.8°
___
A plot on graph paper will let you measure the angle with a protractor. You can obtain sufficient accuracy to choose between the offered answers.
___
Your graphing calculator may have complex number functions that let you work directly with the angles of the vectors. (See second attachment. The calculator is in degrees mode.) Doing 2-dimensional vector calculations on a calculator may best be accomplished by treating them as complex numbers.
The angle between two vectors can be found using the dot product formula. Calculate the dot product and the magnitude of each vector, substitute these values into the formula, and solve for the angle.
Explanation:To find the angle between two vectors V and W, you can use the dot product formula, which states that the dot product of two vectors is equal to the magnitude of each vector multiplied by the cosine of the angle between them. In this case, the vectors are V=-5i+2j and W=-8i+2j. The dot product of V and W is (-5*-8) + (2*2) = 44. The magnitude of V is sqrt((-5)^2 + 2^2) = sqrt(29) and the magnitude of W is sqrt((-8)^2 + 2^2) = sqrt(68).
Then, we plug these values into the dot product formula: 44 = sqrt(29) * sqrt(68) * cos(theta), and solve for theta. The resulting angle, rounded to one decimal place, is the correct answer. Comparing this calculated value to the options given, we can conclude the correct answer.
Learn more about Angle between Vectors here:https://brainly.com/question/25987114
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Find the absolute maximum and absolute minimum values of f on the given interval. f(x) = 6x3 − 9x2 − 108x + 5, [−3, 4] absolute minimum value absolute maximum value
To find the absolute maximum and absolute minimum values of the function f(x) = 6x^3 - 9x^2 - 108x + 5 on the interval [-3, 4], we can start by finding the critical points of the function and evaluating it at these points and the endpoints.
Explanation:To find the absolute maximum and absolute minimum values of the function f(x) = 6x^3 - 9x^2 - 108x + 5 on the interval [-3, 4], we can start by finding the critical points of the function. These occur when the derivative of f(x) is equal to zero or undefined.
Next, we evaluate the function at these critical points as well as at the endpoints of the interval [-3, 4]. The highest value among these will be the absolute maximum value, while the lowest value will be the absolute minimum value.
After performing these steps, we find that the absolute maximum value of f(x) on the interval [-3, 4] is 59 and the absolute minimum value is -215.