Answers
Answer:
[tex]f(x)=\left\{ \begin{array}{rcl}\dfrac{8}{3}x+4&\text{for}&x\le3\\\\-3(x-3)^2+12&\text{for}&x>3\end{array} \right.[/tex]
Step-by-step explanation:
The line to the left of x=3 goes up from 4 to 12 as x goes from 0 to 3. Thus, the slope of it is ...
slope = (12 -4)/(3 -0) = 8/3
It intersects the y-axis at y=4, so its equation is ...
y = (8/3)x +4
__
For x > 3, we observe that the curve falls 3 units (from 12 to 9) as x goes from 3 to 4, then falls 9 units (from 9 to 0) as x goes from 4 to 5. The slope of the curved portion at x=3 looks like it might be zero, suggesting a polynomial function instead of an exponential function.
We note that the change in the value of y=x^2 as x goes from 0 to 1 is 1, then as x goes from 1 to 2, it is 4-1 = 3 — 3 times the change in the first interval. This suggests a quadratic function that has been scaled by a vertical factor of -3, and had its vertex moved to (3, 12). Such a function is described by ...
y = -3(x -3)² +12
Then the graph is modeled by a piecewise function, defined as a line for x < 3, and as a quadratic curve for x > 3. Since the function is continuous at x=3, we can put "or equal to" signs on either or both of these boundaries. We choose to write it as ...
f(x) = (8/3)x +4, x ≤ 3; -3(x -3)² +12, x > 3.
____
The graph of our function is attached. It substantially matches the given graph.
Related Questions
Someone please help me with this-angles of elevation and depression
Answers
Answer:
11.2 miles
Step-by-step explanation:
The mnemonic SOH CAH TOA reminds you that ...
Tan = Opposite/Adjacent
The side you are given is adjacent to the given angle, and the side you are asked to find is opposite the given angle. You can fill in the given values and solve for the required measure.
tan(66°) = BC/(5 mi)
Multiplying by 5 mi gives ...
BC = (5 mi)·tan(66°) = (5 mi)·2.246 = 11.23 mi ≈ 11.2 mi
The area of a rectangular field is 4284 yd. If the width of the field is 51 yards, what is its length?
Answers
Answer:
The length of the field is 84 yards
Step-by-step explanation:
Since the area of a rectangle is width times the length, we can write a simple equation.
Let w be the width
Let l be the length
W*L = 4284
Since we know the width of the rectangle is 51 yards, we can plug it in.
51L = 4284
Dividing 51 on both sides,
L = 84
PLEASE HELP!!!
Consider the function f(x) = 3x and a translation of f(x) named g(x). Janeen created a table for select values of x. Her table is shown below. What can be concluded about the relationship between the two functions? Check all that apply.
The functions have the same base.
The functions have the same range.
The functions have the same exponent.
The functions have the same domain.
g(x) is a translation left 1 unit.
g(x) is a translation right 2 units.
g(x) is a translation up 2 units.
Answers
Answer:
The correct options are:
So, these functions have the same range.
The functions have the same base.
The functions have the same domain.
g(x) is a translation left 1 unit.
Step-by-step explanation:
According to the table, both are exponential functions.
We have that
[tex]f(x) = 3^{x}[/tex]
[tex]g(x) = 3^{x+1}[/tex]
Lets see each affirmation:
The functions have the same base.
An exponential function [tex]a^{x}[/tex] has base a.
In this problems, both f and g have base 3.
The functions have the same range.
The range of f are all the values that f can assume. That is, all the positive numbers.
The range of g are all the values that g can assume. That is, also all the positive numbers.
So, these functions have the same range.
The functions have the same exponent.
An exponential function [tex]a^{x}[/tex] has exponent x.
f has exponent x and g has exponent x + 1. So those functions do not have the same exponent.
The functions have the same domain.
Yes, they both have x = {0,1,2,3} as domain.
g(x) is a translation left 1 unit.
g(x) = f(x+1). So yes, g(x) is a translation left 1 unit.
g(x) is a translation right 2 units.
g(x) is not f(x-2). So g(x) is not a translation right 2 units.
g(x) is a translation up 2 units.
g(x) is not f(x) + 2. So g(x) is not a translation up 2 units.
Answer:
A B D E
Step-by-step explanation:
Just did it!
The temperature at the ski slope was ˉ3°C in the morning. The temperature dropped 9°C by 7 p.m.
What was the temperature at 7 p.m.?
°C
Answers
Answer:
12ºC
Step-by-step explanation:
Morning = -3ºC
At 7pm, temperature dropped ny 9ºC.
Temperature = -3 - 9 = -12ºC
one days discharge at its mouth, 3 trillion gallons, could supply all of country A’s household for 5 months. how much water an average household uses each month. assume there are 200 million households
Answers
let's use engineering notation for the sake of brevity.
1 trillion is 1,000,000,000,000, or just 1E12, twelve zeros.
1 million is then 1E6, six zeros.
we know the discharge for one day is 3E12 gallons, and that'd do just fine for country A for 5 months. How many gallons in 1 month only?
[tex]\bf \begin{array}{ccll} gallons&months\\ \cline{1-2} 3E12&5\\ x&1 \end{array}\implies \cfrac{3E12}{x}=\cfrac{5}{1}\implies 3E12=5x \\\\\\ \cfrac{3E12}{5}=x\implies 6E11=x\implies 600000000000=x[/tex]
if there are 200million inhabitants in A, namely 200E6 or 2E8 inhabitants, how many gallons per inhabitant from all those 6E11 gallons?
[tex]\bf \begin{array}{ccll} gallons&households\\ \cline{1-2} 6E11&2E8\\ x&1 \end{array}\implies \cfrac{6E11}{x}=\cfrac{2E8}{1}\implies 6E11=2E8x \\\\\\ \cfrac{6E11}{2E8}=x\implies \cfrac{600000000000}{200000000}=x\implies 3000=x[/tex]
Which angle appears to be acute?
Answers
Answer:
It is the last one. It is clearly less than 90 degrees.
Step-by-step explanation:
HELP You roll a number cube and flip a coin. What is the probability of rolling a number greater than 1 and flipping heads? Write your answer as a mixed number. WIll give points
Answers
Answer:
⁵/₁₂
Step-by-step explanation:
1. The probability or rolling a number > 1 (given all sides have equal chances):
Possible outcomes: 1, 2, 3, 4, 5, 6; Wanted outcomes: 2, 3, 4, 5, 6
5/6 of the possible outcomes are wanted.
2. Chance of flipping heads (given both sides have an equal chance):
Possible outcomes: heads, tails; Wanted outcomes: heads
1/2 of the possible outcomes are wanted.
3. Probability of both: ⁵/₆ * ¹/₂ = ⁵/₁₂
HELP PLSS WILL GIVE BRAINLIEST AND 99 POINTS
A bag contains fifty blue and fifty green marbles. Two marbles at a time are randomly selected. If both are green, they are placed in box A; if both are blue, in box B; if one is green and the other is blue, in box C. After all marbles are drawn, what is the probability that the number of marbles in box A and box B are the same?
Answers
Answer:
the blue and green marbles are evenly matched, fifty and fifty
each time a single colored pair is drawn, there is now an "extra" unpaired marble of the other color this extra marble must eventually be drawn along with one of the same color
the number of marbles in the A and B boxes can be anywhere from zero to fifty, but it is the SAME for both boxes
9. A 5,000 kg train is traveling at a velocity of 100 m/s and hits another train. The two trains stick together, and the new velocity is 50 m/s. What is the mass of the second train? A. 15,000 kg B. 8,000 kg C. 10,000 kg D. 5,000 kg
Answers
Answer:
D. 5000 kg
Step-by-step explanation:
We assume the second train was standing still and that momentum is conserved. The the product of mass and velocity before the collision is
(5000 kg)·(100 m/s) = 500,000 kg·m/s.
After the collision, where M is the mass of the second train, the momentum is ...
((5000+M) kg)·(50 m/s) = 500,000 kg·m/s
Dividing by 50 m/s and subtracting 5000 kg, we have ...
(5000 +M) kg = 10,000 kg
M kg = 5000 kg
The mass of the second train is 5000 kg.
Answer:
D. 5000 kg
Step-by-step explanation:
The table shows the relationship y=kx.
what is the constant of proportionality,k?
x, 60,45,30,15
y, 12,9,6,3
A. 5/3
B.3/5
C.1/5
D.1/3
Answers
Answer:
the answer is 1/5 because y/x gives you k
The constant of proportionality [tex]k[/tex] is [tex]\frac{1}{5}[/tex].
What is constant of proportionality.
The constant of proportionality is the constant value of the ratio between two proportional quantities.
[tex]$k=\frac{y}{x}$[/tex]
Where [tex]k[/tex] is the constant of proportionality.
It is given that the table which shows the relationship [tex]y=kx[/tex] is,
x y
60 12
45 9
30 6
15 3
We have to find the value of [tex]k[/tex].
According to the formula of constant of proportionality,
Let,
[tex]${{k}_{1}}=\frac{{{y}_{1}}}{{{x}_{1}}}$[/tex]
[tex]${{k}_{1}}=\frac{12}{60}[/tex]
[tex]$\therefore {{k}_{1}}=\frac{1}{5}$[/tex]
Now,
Let,
[tex]${{k}_{2}}=\frac{{{y}_{2}}}{{{x}_{2}}}$[/tex]
[tex]${{k}_{2}}=\frac{9}{45}[/tex]
[tex]$\therefore {{k}_{2}}=\frac{1}{5}$[/tex]
The third is,
[tex]${{k}_{3}}=\frac{6}{30}[/tex]
[tex]$\therefore {{k}_{3}}=\frac{1}{5}$[/tex]
and
[tex]${{k}_{4}}=\frac{3}{15}[/tex]
[tex]$\therefore {{k}_{4}}=\frac{1}{5}$[/tex]
Hence, the Option [tex]$C$[/tex] is correct answer.
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The table shows the values of a function f(x).
What is the average rate of change of f(x) from -2 to 2?
Answers
Answer:
The average rate of change is [tex]=-4[/tex]
Step-by-step explanation:
The average rate of change from x=a to x=b of f(x) is given by;
[tex]=\frac{f(b)-f(a)}{b-a}[/tex]
We want to find the average rate of change of the function represented in the table from x=-2 to x=2.
[tex]=\frac{f(2)-f(-2)}{2--2}[/tex]
From the table f(-2)=25 and f(2)=9
The average rate of change from x=-2 to x=2 is
[tex]=\frac{9-25}{2--2}[/tex]
[tex]=\frac{-16}{4}[/tex]
[tex]=-4[/tex]
Answer:
-4
Step-by-step explanation:
help please
must show work
problem 3 and problem 5
Answers
Answer:
#3. (x-7)(x-4) #5. (2k+7)(k-2)
Step-by-step explanation:
For #3 you have to first get all the terms on one side of the equals sign and set it equal to 0. So that gives us
[tex]x^{2} -11x+28=0[/tex].
Our a value is 1 and our c value is 28, so the product of those is 28. Find the factors of 28 and the combination of those factors that add up to equal the linear term -11x is the combination we need for our problem. The factors of 28 are: 1, 28; 2, 14; 4, 7. 4 and 7 give us 11 when we add them, but since we need a -11, we have to use the negative of both the factors since -7 + -4 = -11. Set up your equation now using the -7 and the -4, "larger" number first (the absolute value which makes the 7 larger):
[tex]x^{2} -7x-4x+28[/tex]. This is the pattern that you will use to factor the next problem, as well.
Group the terms in groups of 2 to get:
[tex](x^{2} -7x)-(4x-28)=0[/tex]. Notice the sign change in front of the 28 in the second set of parenthesis. This is because if I distribute the negative infront of the parenthesis back in, negative times a negative will give us the +28 in the original problem. The same will apply again in #5 when we get there.
Now factor out whatever is common from each set of parenthesis:
[tex]x(x-7)-4(x-7)=0[/tex].
Now the common term is the factor (x-7) so that can be factored out now, leaving behind:
(x-7)(x-4). That's the answer for #3.
Now for #5:
We will start by getting everything on one side (I am changing the k's to x's):
[tex]2x^{2} +3x-14=0[/tex].
The product of our a value and c value is again 28. Find the combination of the factors of 28 that will add to give us the middle (linear) term of 3: That is again 7 and 4, with the 7 needing to be positive and the 4 needing to be negative since 7 - 4 = 3. Set up our expanded quadratic as follows, "larger" number (the absolute value of) first:
[tex]2x^{2} +7x-4x-14=0[/tex].
Group them into groups of 2 again:
[tex](2x^{2} +7x)-(4x+14)=0[/tex]
Again, notice the necessary sign change so when we distribute the negative back into the parenthesis we get the -14 we started with in the original problem.
Now factor out what is common from each set of parenthesis:
[tex]x(2x+7)-2(2x+7)=0[/tex].
What's common now is the factor (2x+7) so that can be factored out leaving behind
(2x+7)(x-2)=0
And you're done!!!
6)
Q:
A:
Lynn has one red, one blue, and one yellow shirt to choose from. She also has one blue and one yellow necklace to choose from. Lynn chooses a random shirt and necklace for work today. Are these events dependent or independent? Give a reason.
A) Dependent. These primary colors mix together to make secondary colors.
B) Dependent. After Lynn chooses a shirt, there are only two more shirts to choose from.
C) Independent. The wardrobe is not large enough for one thing to affect something else.
D) Independent. Randomly selecting a shirt does not affect randomly selecting a necklace.
Answers
i would pick D as the answer.
The events of Lynn choosing a shirt and choosing a necklace are independent. The choice of one item does not affect the choice of the other. The correct answer is D) Independent - Randomly selecting a shirt does not affect randomly selecting a necklace.
Explanation:The question asks whether the events of Lynn choosing a shirt and necklace are dependent or independent. In general, events are called independent if the outcome of one does not affect the outcome of another. In this scenario, picking a shirt does not change the options of necklaces and vice versa. Thus, the selection of a shirt has no effect on the selection of a necklace.
The correct answer is therefore: Independent. Randomly selecting a shirt does not affect randomly selecting a necklace. Even if Lynn picked a red, blue, or yellow shirt first, it wouldn't affect her choice of necklace. She will still have the same two necklaces to choose from regardless of what shirt she picks.
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a tree 24 feet tall casts a shadow 12 feet long. Brad is 6 feet tall. How long is Brad's shadow? draw a diagram and solve
Answers
Answer:
3 feet long
Step-by-step explanation:
Answer: Brad's shadow is 3 feet long
Which of the following correctly describes the graph of this function?
Answers
Answer:
A. The graph of the function increases and decreases over its domain.
Step-by-step explanation:
The graph is attached. Some places, it has positive slope (is increasing); other places it has negative slope (is decreasing).
Jane had 275 stickers, which she gave to her friends. Each friend got five more stickers than the friend before. If the first friend got 5 stickers, how many friends got stickers from Jane?
Answers
Answer:
10 friends
Step-by-step explanation:
we know that
The formula of the sum is equal to
[tex]sum=\frac{n}{2}[2a1+(n-1)d][/tex]
where
a1 is the first term
n is the number of terms (number of friends)
d is the common difference in the arithmetic sequence
In this problem we have
[tex]sum=275\ stickers[/tex]
[tex]a1=5\ stickers[/tex]
[tex]d=5[/tex] ----> the common difference
substitute in the formula and solve for n
[tex]275=\frac{n}{2}[2(5)+(n-1)(5)][/tex]
[tex]550=n[10+5n-5]\\ \\550=10n+5n^{2} -5n\\ \\5n^{2}+5n-550=0[/tex]
Solve the quadratic equation by graphing
The solution is n=10
see the attached figure
therefore
She had 10 friends who got stickers
Evaluate the integral Integral from left parenthesis 2 comma 1 comma 2 right parenthesis to left parenthesis 6 comma 7 comma negative 5 right parenthesis y dx plus x dy plus 7 dz by finding parametric equations for the line segment from (2,1,2) to (6,7,negative 5) and evaluating the line integral of Fequalsyiplusxjplus7k along the segment. Since F is conservative, the integral is independent of the path.
Answers
To solve this integral, one would establish parametric equations for the line segment between the points given, substitute the parametric equations into the vector field, and then compute the integral.
Explanation:To solve the integral Integral from (2, 1, 2) to (6, 7, -5) y dx + x dy + 7 dz, it is necessary to establish the parametric equations for the line. The equation for a path in a three-dimensional space between two points can be defined as: r(t) = (1 - t)A + tB, where A and B represent the initial and final points respectively, and t is the parameter that varies between 0 and 1.
For our case the points A = (2, 1, 2) and B = (6, 7, -5), so r(t) would become r(t) = (1 - t)(2, 1, 2) + t(6, 7, -5) = (2 + 4t, 1 + 6t, 2 - 7t).
Subsequently, you need to calculate the line integral of F along this path. Here, F = yi + xj + 7k, and when the values of x, y, and z from the function r(t) are substituted into F, the result is F = (1 + 6t)i + (2 + 4t)j + 7k.
Finally, evaluate the line integral. Due to the conservative nature of the field, the value of the integral would be the same over any path leading from (2,1,2) to (6,7,-5)
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This answer calculates a line integral of a conservative vector field using the given parametric equations. The answer lies in plugging the parametric equations into the integral and calculating it out.
Explanation:This question involves calculating a line integral for a vector field. The vector field represented here is F = yi + xj + 7k. Parametric equations for the line segment from (2,1,2) to (6,7,-5) can be written as x = 2 + 4t, y = 1 + 6t, z = 2 -7t with 0 ≤ t ≤ 1.
Next, we plug into the integral ∫F.dr from t=0 to t=1, where dx = 4dt, dy = 6dt and dz = -7dt. This gives us the integral ∫_{0}^{1} (ydx + xdy + 7dz). Inserting the parametric equations into this integral and calculating it out gives the value of the line integral.
This procedure embodies the idea that conservative vector fields have the property that the line integral is independent of the path, meaning the value of the integral depends only on the endpoints of the path and not on the specific path taken.
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You are framing a picture with a frame of equal width on each side. Find the perimeter and the area of the picture including the frame when the width of the frame is 2 inches. The length is 20 inches and the width is 16 inches. The width of the frame is 2 inches. what is a polynomial for the area and the perimeter. Also what is the perimeter and the area of the picture including the frame when the width of the frame is 2 inches.
Answers
Answer:
Part a) The polynomial for the area including the frame is
[tex](4x^{2}+72x+320)\ ft^{2}[/tex]
Part b) The polynomial for the perimeter including the frame is
[tex](72+8x)\ ft[/tex]
Part c) The perimeter of the picture including the frame when the width of the frame is 2 inches is equal to [tex]88\ ft[/tex]
Part d) The area of the picture including the frame when the width of the frame is 2 inches is equal to [tex]480\ ft^{2}[/tex]
Step-by-step explanation:
Let
x-----> the width of the frame
L-----> the length of the picture
W-----> the width of the picture
Part a) What is a polynomial for the area including the frame?
we have
The dimensions of the picture are
[tex]L=20\ in[/tex]
[tex]W=16\ in[/tex]
The area including the frame is equal to
[tex]A=(20+2x)(16+2x)\\ \\A=320+40x+32x+4x^{2}\\ \\A=(4x^{2}+72x+320)\ ft^{2}[/tex]
Part b) What is a polynomial for the perimeter including the frame?
we have
The dimensions of the picture are
[tex]L=20\ in[/tex]
[tex]W=16\ in[/tex]
The perimeter including the frame is equal to
[tex]P=2[(20+2x)+(16+2x)]\\ \\P=2[36+4x]\\ \\P=(72+8x)\ ft[/tex]
Part c) What is the perimeter of the picture including the frame when the width of the frame is 2 inches
we have
[tex]P=(72+8x)\ ft[/tex]
For x=2 in
substitute
[tex]P=72+8(2)=88\ ft[/tex]
Part d) What is the area of the picture including the frame when the width of the frame is 2 inches
we have
[tex]A=(4x^{2}+72x+320)\ ft^{2}[/tex]
For x=2 in
substitute
[tex]A=(4(2)^{2}+72(2)+320)=480\ ft^{2}[/tex]
The perimeter of a two-dimensional figure is the distance covered around it.
The polynomial for the area including the frame is [tex]\rm 4x^2+72x +320[/tex].
The polynomial for the perimeter including frame is [tex]\rm 72+8x[/tex].
The perimeter of the picture including the frame when the width of the frame is 2 inches is 88 feet.
The area of the picture including the frame when the width of the frame is 2 inches is 480 feet.
Given thatYou are framing a picture with a frame of equal width on each side.
The width of the frame is 2 inches.
The length is 20 inches and the width is 16 inches.
The width of the frame is 2 inches.
What is the perimeter?The perimeter of a two-dimensional figure is the distance covered around it.
Let the width of the frame be x.
The length of the picture be L.
The width of the picture is W.
1. What is a polynomial for the area including the frame?
[tex]\rm Area \ of \ the \ frame = length \times width\\\\ Area \ of \ the \ frame = (20+2x) (16+2x)\\\\ Area \ of \ the \ frame = 320+40x+32x+4x^2\\\\ Area \ of \ the \ frame = 4x^2+72x +320[/tex]
The polynomial for the area including the frame is [tex]\rm 4x^2+72x +320[/tex].
2. What is a polynomial for the perimeter including the frame?
[tex]\rm Perimeter \ of \ the \ picture = 2 (length + width)\\\\ Perimeter \ of \ the \ picture = 2(20+2x+16+2x)\\\\ Perimeter \ of \ the \ picture = 2(36+4x)\\\\ Perimeter \ of \ the \ picture = 72+8x[/tex]
The polynomial for the perimeter including frame is [tex]\rm 72+8x[/tex].
3. What is the perimeter of the picture including the frame when the width of the frame is 2 inches.
[tex]\rm Perimeter \ of \ the \ picture = 72+8x\\\\Perimeter \ of \ the \ picture = 72+8(2)\\\\ Perimeter \ of \ the \ picture = 72+16\\\\Perimeter \ of \ the \ picture = 88\\\\[/tex]
The perimeter of the picture including the frame when the width of the frame is 2 inches is 88 feet.
4. What is the area of the picture including the frame when the width of the frame is 2 inches.
[tex]\rm Area \ of \ the \ frame = 4x^2+72x +320\\\\ Area \ of \ the \ frame = 4(2)^2+72(2)+320\\\\ Area \ of \ the \ frame=480[/tex]
The area of the picture including the frame when the width of the frame is 2 inches is 480 feet.
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A cone is formed by rotating △ABC around segment BC . Which expression can be used to find the volume of the cone?
Answers
Answer:
[tex]\pi\dfrac{12.1}{3}\left(\dfrac{12.1}{\tan 57^{\circ}}\right)^2[/tex]
Step-by-step explanation:
You are interested in two of the sides of the right triangle, the leg AC and the leg BC. The trig function that relates their values to the angle shown is told you by the mnemonic SOH CAH TOA, which reminds you ...
Tan = Opposite/Adjacent
The side opposite the angle, BC is given as 12.1; the side adjacent is AC, designated r. Then the above relation tells you ...
tan(57°) = 12.1/r
Rearranging, we have ...
r = 12.1/tan(57°)
The volume of the cone is given by the formula ...
V = (1/3)πr²h . . . . . where h = 12.1
Filling in what we know, this is ...
V = (1/3)π(12.1/tan(57°))²·12.1
This can be rearranged to the form shown in your answer choices:
V = π(12.1/3)(12.1/tan(57°))² . . . . . . matches the lower-right choice
Answer:
a.
Step-by-step explanation:
Use Stokes' Theorem to evaluate C F · dr where C is oriented counterclockwise as viewed from above. F(x, y, z) = i + (x + yz)j + (xy − sqrt(z )k, C is the boundary of the part of the plane 7x + 6y + z = 1 in the first octant.
Answers
By Stokes' theorem, the line integral of [tex]\vec F[/tex] over [tex]C[/tex] is equivalent to the surface integral of the curl of [tex]\vec F[/tex] over [tex]S[/tex], where [tex]S[/tex] is the part of the plane [tex]7x+6y+z=1[/tex] in the first octant, with [tex]S[/tex] having positive/upward orientation.
Parameterize [tex]S[/tex] by
[tex]\vec s(u,v)=\dfrac{(1-u)(1-v)}7\,\vec\imath+\dfrac{u(1-v)}6\,\vec\jmath+v\,\vec k[/tex]
with [tex]0\le u\le1[/tex] and [tex]0\le v\le1[/tex].
Take the normal vector to [tex]S[/tex] to be
[tex]\vec s_u\times\vec s_v=\dfrac{1-v}6\,\vec\imath+\dfrac{1-v}7\,\vec\jmath+\dfrac{1-v}{42}\,\vec k[/tex]
The curl of [tex]\vec F[/tex] is
[tex]\nabla\times\vec F(x,y,z)=(x-y)\,\vec\imath-y\,\vec\jmath+\vec k[/tex]
Then the line integral is equivalent to
[tex]\displaystyle\int_C\vec F\cdot\mathrm d\vec r=\iint_S(\nabla\times\vec F)\cdot\mathrm d\vec S[/tex]
[tex]=\displaystyle\int_0^1\int_0^1\left(\frac{(6-13u)(1-v)}{42}\,\vec\imath-\dfrac{u(1-v)}6\,\vec\jmath+\vec k\right)\cdot\left(\dfrac{1-v}6\,\vec\imath+\dfrac{1-v}7\,\vec\jmath+\dfrac{1-v}{42}\,\vec k\right)\,\mathrm du\,\mathrm dv[/tex]
[tex]=\displaystyle\frac1{252}\int_0^1\int_0^1(12-6v-19u+19uv)(1-v)\,\mathrm du\,\mathrm dv=\boxed{\frac{11}{1512}}[/tex]
First, find the normal vector of the plane, and then find the curl of the vector field to obtain a new vector field to be integrated over the given surface. Then perform the surface integral of this resultant vector field over the surface to get the required output.
Explanation:Using Stokes' Theorem, we first find the normal vector of the plane 7x + 6y + z = 1 by extracting the coefficients, i.e., the vector is (7,6,1). Next, find the curl of the vector field F(x, y, z) = i + (x + yz)j + (xy - sqrt(z ))k. This provides the vector field, let's say G, that you are going to integrate over the surface that C bounds. The curl is calculated as follows:
The curl of F = ∇ x F= ( ∂/∂y [(xy - sqrt(z))] - ∂/∂z [(x + yz)] )i - ∂/∂x [(xy - sqrt(z))]j + ∂/∂x [(x + yz)]k.
Then perform the surface integral of this resultant vector field, G, over the surface bounded by C to get your answer. The surface integral is ∫∫ G·dS.
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If Joanne divides 4.4 E-10 by 5.5 E17 using a calculator, she should get the answer
__E____?
Answers
Answer:
8 E -28
Step-by-step explanation:
Use the 2nd button and the comma button to get the "EE" which puts numbers into scientific notation for you. I use parenthesis and this is what my screen looks like right before I hit "enter" to get the answer:
[tex](4.4 E-10)/(5.5E17)[/tex]
Hit enter and you get 8 E -28
Answer:
8E -28
.
Step-by-step explanation:
What is the equation of the line that is perpendicular to the given line and passes through the point (2, 6)?
x = 2
x = 6
y = 2
y = 6
Answers
Answer:
x=2
Step-by-step explanation:
Thanks for helping me out
Answers
Answer:
0.3
Step-by-step explanation:
To calculate the dilation factor of two similar figures, you simply need to establish the ratio of the dilated figure (in our case PQRS) to the original figure (ABCD) using the same relative side.
In our case, we'll take the longest of the sides provided in each figure.
So, we divide 12.6 by 42 to know the dilation factor of the longest side, it will be the same for all other sides:
d = 12.6 / 42 = 0.3
Suppose the required reserve ratio is 20 percent. A $5 million deposit allows commercial banks to create as much as
a) $25 million.
b) $5 million.
c) $10 million.
d) $1 million.
Answers
Answer:
Option D. $1 million
Step-by-step explanation:
we know that
Reserve Ratio, it is the percentage of deposits which commercial banks are required to keep as cash
Find the 20% of $5 million
20%=20/100=0.20
0.20*5,000,000=$1,000,000
so
$1 million
The required reserve ratio is what banks must keep from a deposit. Given a 20% reserve ratio and a $5 million deposit, banks must reserve $1 million. Using the money multiplier formula, the remaining funds could theoretically create as much as $25 million in money supply.
Explanation:The required reserve ratio is the percentage of deposits that a bank must hold as reserves. In this case, the required reserve ratio is 20 percent. The rest of the deposit (80 percent) can be loaned out or invested by the bank, which will create additional deposits and thus increase the money supply.
If a commercial bank receives a $5 million deposit and the required reserve ratio is 20 percent, the bank must keep $1 million (20 percent of $5 million) as reserve. The rest, $4 million, can be loaned out or invested.
However, this doesn't simply stop at $4 million. This loaned money will eventually be deposited back into the banking system (let's assume to another bank), which can then loan out 80% of that deposited money, and this cycle can continue. In an extremely simplified scenario, you can continue this process until the banks can no longer lend out money.
To simplify this scenario, the money multiplier formula can be used. The money multiplier formula is 1 divided by the reserve ratio. In this case, it's 1 / 0.20 = 5. Therefore, a $5 million deposit can potentially lead to a $25 million increase in the money supply, so the answer is (a) $25 million.
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Please help!! 20 POINTS
Answers
The answers are:
First image:
The answer is the second option, the angles is [tex]53\°[/tex]
Second image:
The answer is the third option:
[tex]\frac{5}{13}[/tex]
Third image:
The length of the adjacent leg is the first option:
[tex]8\sqrt{2}units[/tex]
Fourth image:
The answer is the fourth option, [tex]72\°[/tex]
Fifth image:
The answer is the fourth option, DF (hypothenuse) is equal to 25 units.
Why?To solve these problems, we need to use the following trigonometric identities and the Pythagorean Theorem, since we are working with right triangles.
[tex]Tan(\alpha)=\frac{y}{x}\\\\(Tan(\alpha))^{-1} =(\frac{y}{x})^{-1}\\\\\alpha =Arctan(\frac{y}{x})[/tex]
[tex]Sin(\alpha)=\frac{opposite}{hypothenuse}[/tex]
Pythagorean Theorem:
[tex]c^{2}=a^{2} +b^{2}[/tex]
So, solving we have:
First image:
We are given a right triangle that has the following lengths:
[tex]base=x=6units\\height=y=8units\\hypothenuse=10units[/tex]
Then, calculating we have:
[tex]\alpha =Arctan(\frac{y}{x})\\\\\alpha =Arctan(\frac{8}{6})\\\\\alpha =Arctan(1.33)\\\\\alpha =53\°[/tex]
Hence, the answer is the second option, the angles is [tex]53\°[/tex]
Second image:
We are given a right triangle that has the following lengths:
[tex]base=x=12units\\height=y=5units\\hypothenuse=13units[/tex]
Then calculating the sin ratio, we have:
[tex]Sin(\alpha)=\frac{opposite}{hypothenuse}[/tex]
[tex]Sin(\alpha)=\frac{5}{13}[/tex]
Thence, the answer is the third option:
[tex]\frac{5}{13}[/tex]
Third Image:
We are given the following information:
[tex]hypothenuse=16units\\\\\alpha =45\°[/tex]
Then, calculating one of the angle legs, since both will have the same length, using the sine trigonometric identity, we have:
[tex]Sin(\alpha)=\frac{Opposite}{Hypothenuse}\\ \\Sin(45\°)=\frac{Opposite}{16}\\\\Opposite=Sin(45\°)*16\\\\Opposite=\frac{\sqrt{2} }{2}*16=8\sqrt{2}[/tex]
Hence, the answer is the first option the length of the adjacent leg is
[tex]Opposite=\frac{\sqrt{2} }{2}*16=8\sqrt{2}units[/tex]
Fourth image:
We are given the following information:
[tex]base=x=9units\\height=y=3units[/tex]
To calculate the angle at the B vertex, first, we need to calculate the angle at the C vertex, and then, calculate the B vertex by the following way:
Since the sum of all the interior angles of a triangle are equal to 180°, we have that:
[tex]180\°=Angle_{B}+Angle{C}+90\°[/tex]
[tex]Angle_{B}=180\° -90\°-Angle_{C}[/tex]
So, calculating the angle at the C vertex, we have:
[tex]\alpha =Arctan(\frac{y}{x})[/tex]
[tex]\alpha =Arctan(\frac{3}{9})[/tex]
[tex]\alpha =Arctan(0.33)=18.26\°[/tex]
Then, calculating the angle at the B vertex, we have:
[tex]Angle_{B}=180\° -90\°-18.26\°=71.74\°=71.8\°=72\°[/tex]
Hence, the answer is the fourth option, [tex]72\°[/tex]
Fifth image:
We are given the following information:
[tex]base=x=24units\\height=y=7units[/tex]
Now, to calculate the distance DF (hypothenuse) we need to use the Pythagorean Theorem:
[tex]c^{2}=a^{2} +b^{2} \\\\hypothenuse^{2}=adjacent^{2}+opposite^{2}\\\\\sqrt{hypothenuse^{2}}=\sqrt{adjacent^{2}+opposite^{2}}\\\\hypothenuse=\sqrt{adjacent^{2}+opposite^{2}}[/tex]
Then, substituting we have:
[tex]hypothenuse=\sqrt{24^{2}+(7)^{2}}[/tex]
[tex]hypothenuse=\sqrt{576+49}=\sqrt{625}[/tex]
[tex]hypothenuse=\sqrt{625}[/tex]
[tex]hypothenuse=25units[/tex]
Hence, the answer is the fourth option, DF (hypothenuse) is equal to 25 units.
Have a nice day!
A fish tank is the shape of a cube. It’s volume is 125ft. What is the area of one side of the tank
Answers
Answer:
The area of one side of the tank is [tex]25ft^{2}[/tex]
Step-by-step explanation:
A cube is a regular Polyhedron limited by six equal squares. It is also known as the regular hexahedron, or simply hexahedron.
The cube is a ortohedron, that is, it has its six faces straight and perpendicular to each other. In addition to being their rectangle faces, they are also square.
It has the following properties:
*Number of faces: 6
*Number of vertices: 6
*Number of edges: 12
*Number of edges from one vertex: 3
Being the length of the edge of the cube, then its Volume (V) is expressed by the formula: [tex]v=a^{3}[/tex] Being a the length of the edge of the cube
A fish tank is the shape of a cube. Its volume is [tex]125ft^{3}[/tex], to calculate the are of one side of the tank, we know that each faces has the form of a square of area [tex]a^{2}[/tex].
To calculate the value of a.
[tex]a=\sqrt[3]{125ft^{3} } =5ft[/tex]
Then the area of square is [tex]a^{2}[/tex]
Substituting the value of a
[tex](5ft)^{2} =25ft^{2}[/tex]
Answer:
25 feet squared
Step-by-step explanation:
50 is what percent of 32?
Answers
Answer:
156.25% =P
Step-by-step explanation:
Is means equals and of means multiply
50 = P * 32
Divide each side by 32
50/32 = 32P/32
1.5625= P
Now we need to change it to percent form by multiplying by 100%
1.5625 * 100% = P
156.25% =P
Which of the following has a graph that is a straight line? Equation 1: y = 5x2 + 41 Equation 2: y = 14x5 − 4 Equation 3: y = 12x + 17 Equation 4: y4 = 2x − 1
Answers
Answer:
Equation 3: y = 12x + 17
Step-by-step explanation:
Only a linear equation (degree 1) will have a graph that is a straight line.
Equation 1: degree 2Equation 2: degree 5Equation 3: degree 1Equation 4: degree 4The degree of the equation is the highest power of any of the variables. If different variables are in the same term, it is the sum of the powers of those variables.
Answer:
Equation 3: y = 12x + 17
Step-by-step explanation:
It is the answer because all of the variables have an exponent of 1.
Which statement BEST compares the growth of the two functions as x approaches infinity?
Answers
Answer:
D)
Step-by-step explanation:
Please Assist With This:
Today's Beginning Balance: $27.84
You walked and fed a neighbor's dog while the neighbor was on vacation. He returned today and paid you $15.00. You deposited the check into your account. Your friend Caleb invited you to his birthday party. Using your debit card, you spent $10.45 on his gift at the Science Discovery Center.
*Please Show Work
Answers
Answer:
32.39
Step-by-step explanation:
27.84+15=42.84
42.84-10.45= THE ANSWER