To solve this problem, we can set up a system of equations to represent the given information. Solving this system of equations, we find that each small candle holder holds 2 candles and each large candle holder holds 8 candles.
Explanation:To solve this problem, let's represent the number of candles each small candle holder holds as x and the number of candles each large candle holder holds as y. We can set up a system of equations to represent the given information:
For the east side: 10x + 4y = 52For the west side: 2x + 4y = 36We can solve this system of equations by eliminating one variable. Subtracting the equations, we get 8x = 16. Dividing both sides by 8, we find that x = 2. Substituting this value back into one of the equations, we can solve for y. Using the first equation, we have 10(2) + 4y = 52, which simplifies to 20 + 4y = 52. Subtracting 20 from both sides, we obtain 4y = 32. Dividing by 4, we find that y = 8. Therefore, each small candle holder holds 2 candles and each large candle holder holds 8 candles.
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To find out how many candles each candle holder holds, set up a system of equations using the given information and solve for the variables. Each small candle holder holds 2 candles and each large candle holder holds 8 candles.
Explanation:To find out how many candles each candle holder holds, we need to set up a system of equations using the given information. Let's say the number of candles a small candle holder can hold is 's' and the number of candles a large candle holder can hold is 'l'.
From the east side, we have the equation 10s + 4l = 52. From the west side, we have the equation 2s + 4l = 36. Solving these equations, we can find the values of 's' and 'l', which will give us the number of candles each candle holder can hold.
We can eliminate 'l' by multiplying the first equation by 2 and the second equation by 5. This gives us 20s + 8l = 104 and 10s + 20l = 180. By subtracting the second equation from the first, we get 10l = 76. Dividing both sides by 10, we find that l = 7.6. Since we cannot have a fraction of a candle, we can approximate l to the nearest whole number, which is 8.
Substituting the value of l back into one of the original equations, we can find the value of s. Using the first equation, we have 10s + 4(8) = 52. Simplifying this equation gives us 10s + 32 = 52. Subtracting 32 from both sides, we find that 10s = 20. Dividing both sides by 10, we find that s = 2. Therefore, each small candle holder holds 2 candles and each large candle holder holds 8 candles.
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Harper wrote the expression 15-6+7 to represent "15 minus the sum of 6 and7 evaluate 15 -6+7 and then explain why Harper expression is incorrect
Answer:
15-6+7=16. The expression should be written as 15-(6+7).
Step-by-step explanation:
The sum of 6 and 7 is (6+7). 15 minus the sum of 6 and 7 would then be written as 15-(6+7), because this way you subtract the entire sum from 15. The expression 15-6+7 just subtracts 6 from 15, then adds 7 onto this number, which is not the same thing as subtracting 6+7 from 15.
The expression 15 - 6 + 7 does not represent '15 minus the sum of 6 and 7'. The correct expression should be 15 - (6 + 7). Evaluating Harper's expression gives 16, while the correct expression gives 2.
Explanation:The expression 15 - 6 + 7 does not properly relay the concept of '15 minus the sum of 6 and 7'. To correctly express this idea, parentheses should be used: 15 - (6 + 7).
To evaluate Harper's expression, one would operate in the order of their entry: 15 - 6 is 9, then, add 7 to 9 and the result is 16.
However, in the correct expression 15 - (6 + 7), the sum of 6 and 7 equals 13 is computed first, according to the order of operations (PEMDAS/BODMAS). Then, subtract this sum from 15 to get 2.
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The graphs of f(x) and g(x) are shown below:
PIC
What are the solutions to the equation f(x) = g(x)?
x = −3, 4
x = −3.8, −0.4
x = −3.8, 3
x = 6.5, −6
Answer:
-3.8, 3
Step-by-step explanation:
The solution to a system is where the graph cross each other. If you look to where they intersect to the left of the origin, where x is negative, it appears that they intersect ALMOST at -4, but not quite. So -3.8 is going to have to do since we don't have the equations of either graph to find the exact values of x. To the right of the origin, where x is positive, the graphs cross where x = 3..
Answer:
x = −3.8, 3
Step-by-step explanation:
The solutions to f(x) = g(x) are the x-coordinates of the points of intersection of their graphs. Those x-values appear to be about -3.8 and +3.
Line m is parallel to line n. The measure of angle 4 is (5a + 10)°. The measure of angle 6 is (3a + 10)°. What is the measure of angle 4?
A. 110°
B. 70°
C. 20°
D. 60°
Answer:
A. 110
Step-by-step explanation:
Angles 4 and 6 are supplementary so if we add them together they will equal 180.
(5a + 10)° + (3a + 10)° = 180°
Simplify a bit to get 8a + 20 = 180
and 8a = 160.
a = 20. Now sub that value of a into the expression for angle 4:
5a + 10 --> 5(20) + 10 = 110°
Answer:
Option A.
Step-by-step explanation:
Given information: m║n, [tex]m\angle 4=(5a+10)^{\circ}[/tex] and [tex]m\angle 6=(3a+10)^{\circ}[/tex].
If a transversal line intersect two parallel lines, then the interior angles on the same sides are supplementary angles. It means their sum is 180.
From the given figure it is clear that angle 4 angle 6 are interior angles on the same side. So, angle 4 and 6 are supplementary angles.
[tex]m\angle 4+m\angle 6=180^{\circ}[/tex]
[tex](5a+10)+(3a+10)=180[/tex]
On combining like terms we get
[tex](5a+3a)+(10+10)=180[/tex]
[tex]8a+20=180[/tex]
Subtract 20 from both sides.
[tex]8a+20-20=180-20[/tex]
[tex]8a=160[/tex]
Divide both sides by 8.
[tex]a=20[/tex]
The value of a is 20.
[tex]m\angle 4=(5a+10)^{\circ}\Rightarrow 5(20)+10)^{\circ}=110^{\circ}[/tex]
Therefore, the correct option is A.
Which of the following investigations is an example of the study of an abiotic factor?A. the relationship between finch beak size and food availability on two different Galapagos IslandsB. observing interactions among various organisms in a rainforest canopyC. investigating how an elk population competes for foodD. identifying food sources for an egret populationE. investigating how the amount of annual precipitation affects the distribution of a tree species
Answer:
E. Investigating how the amount of annual precipitation affects the distribution of a tree species.
Step-by-step explanation:
Abiotic factors are factors in an ecosystem that are non-living. Abiotic factors include water, soil, temperature, light, and air. Among all the choices, only choice E considers an abiotic factor, which is precipitation.
Precipitation is water, and as mentioned earlier, water is an abiotic factor. The case here is the study of the annual precipitation and effects on distribution of species. The study is specific to the precipitation.
The other choices involve biotic factors, like food source, the organisms and the like.
Note: Remember to show all of the steps that you use to solve the problem. You can use the comments field to explain your work. Your teacher will review each step of your response to ensure you receive proper credit for your answer.
Find all the zeroes of the equation.
–3x4+ 27x2 + 1200 = 0
Answer:
Values of x are 4i, -4i, 5 and -5
Step-by-step explanation:
[tex]3x^4+27x^2 + 1200[/tex] We need to find all the zeros (roots) of the above equation.
Let assume that x^4 = u^2 and x^2 = u
Putting values of x^4 and x^2 in the above equation and finding the value of u.
[tex]-3u^2 + 27u+1200=0\\Using \,\,quadratic\,\,equation\,\,to\,\,solve:\\u=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\\where\,\, a= -3, b= 27 \,\,and\,\, c=1200\\u=\frac{-(27)\pm\sqrt{(27)^2-4(-3)(1200)}}{2(-3)}\\u=\frac{-27\pm\sqrt{729+14,400}}{-6}\\u=\frac{-27\pm\sqrt{729+14,400}}{-6}\\u=\frac{-27\pm\sqrt{15129}}{-6}\\u=\frac{-27\pm123}{-6}\\so, \,\, u = \frac{-27+123}{-6} \,\, and \,\, u \frac{-27-123}{-6}\\u= -16 \,\, and \,\, u = 25[/tex]
So, values of u are -16 and 25
Putting back the value of u i.e, x^2
x^2 = -16 and x^2 =25
solving
Taking square root on both sides:
√x^2 =√-16 and √x^2 = √25
x = ± 4i (as √-1 =i) and x = ±5
So, values of x are 4i, -4i, 5 and -5.
Please help me with this
first use the formula it is 1/3 × base area × height =volume of pyramid
so we just need to replace some of the information given in the statement to our formula
so it's going to be,i'll let the height as unknown X
128=1/3 × (8 × 8) × X
so it's going to be
128=64/3X
so X is 128 ÷ 64/3=6cm
Answer:
I'm Sure the answer is 6 cm.
calculate the value of c
Answer:
14.2
Step-by-step explanation:
For this case, we have to define trigonometric relations in a rectangle triangle that, the tangent of an angle is given by the leg opposite the angle on the leg adayed to it. According to the figure we have:[tex]tg (35) = \frac {c} {5}\\c = tg (35) * 5\\c = 0.70020754 * 5[/tex]
[tex]c = 3.5010377[/tex]
Answer:
Option D
During a hike,3 friends equally shared 1/2 pound of trail mix .What amount of trail mix,in pounds,did each friend receive?
[tex]\bf \cfrac{1}{2}\div 3\implies \cfrac{1}{2}\div \cfrac{3}{1}\implies \cfrac{1}{2}\cdot \cfrac{1}{3}\implies \cfrac{1}{6}[/tex]
3 friends equally shared 1/2 pound of trail mix, during the hike.
Total quantity of trail mix = 1/2 pound
Quantity shared by each friend is 1/3 rd of the trial mix
= 1/3 * 1/2 pound
= 1/6 pound
Therefore, during the hike, each friend shared 1/6 pound of the trail mix.
Hope this helps ..!!
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What is the value of tan theta in the unit circle below? A.1/2 B.sqrt3/3. C.sqrt3/2. D.sqrt3
if you don't have a Unit Circle, this is a good time to get one, you can find many online.
Check the picture below.
Answer:
B . √3/3.
Step-by-step explanation:
Tan theta = opposite side/ adjacent
= 1/2 / √3/2
= 1/2 * 2/√3
= 1 /√3
= √3/3.
Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. g(x) = 7x u2 − 1 u2 + 1 du 6x Hint: 7x f(u) du 6x = 0 f(u) du 6x + 7x f(u) du 0
It looks like you're given
[tex]g(x)=\displaystyle\int_{6x}^{7x}\frac{u^2-1}{u^2+1}\,\mathrm du[/tex]
Then by the additivity of definite integrals this is the same as
[tex]g(x)=\displaystyle\int_0^{7x}\frac{u^2-1}{u^2+1}\,\mathrm du-\int_0^{6x}\frac{u^2-1}{u^2+1}\,\mathrm du[/tex]
(presumably this is what the hint suggests to use)
Then by the fundamental theorem of calculus, we have
[tex]\dfrac{\mathrm dg}{\mathrm dx}=7\dfrac{(7x)^2-1}{(7x)^2+1}-6\dfrac{(6x)^2-1}{(6x)^2+1}=\dfrac{1764x^4+169x^2-1}{1764x^4+85x^2+1}[/tex]
The Fundamental Theorem of Calculus Part 1 states the relationship between differentiation and integration. For a given function, transform the integral as hinted in the question, then use the theorem to differentiate the function by simply replacing the variable of integration with the upper limit of the integral.
Explanation:The Fundamental Theorem of Calculus Part 1 is primarily used to identify the relationship between differentiation and integration, which are two basic operations in calculus. For a given function g(x) = ∫7x u² − 1 u² + 1 du from a to x, we first identify the function inside the integral, let's say f(u) = 7x u² − 1 u² + 1. Now, the mentioned integral transformation hints us that: 7x ∫f(u) du from 6x to 0 equals ∫f(u) du from 6x to 0 + ∫7x f(u) du from 0 to 6x.
Next, we simply differentiate g(x) using Part 1 of the Fundamental Theorem of Calculus, which states that if g(x) is the integral from a to x of f(t) dt, then the derivative of g(x) is f(x). So, by applying this, the derivative function g'(x) of given g(x) will be f(x), i.e., 7x x² − 1 x² + 1.
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If [tex]f(x) = 2x^2+5\sqrt{(x-2)}[/tex] complete the following statement:
f(6) = ____
Answer: 82
Simply pug in 6 for [tex]x[/tex]
[tex]2(6)^2+5\sqrt{(6-2)} \\72+5\sqrt{4} \\72+5(2)\\72+10=\\82[/tex]
Hope this helps and have a great day!!!
[tex]Sofia[/tex]
For this case we have a function of the form [tex]y = f (x).[/tex]
Where:[tex]f (x) = 2x ^ 2 + 5 \sqrt {x-2}[/tex]
We must find the value of the function when x = 6, that is, f (6):
Then, replacing the value of "x"[tex]f (6) = 2 (6) ^ 2 + 5 \sqrt {6-2}\\f (6) = 2 * 36 + 5 \sqrt {4}\\f (6) = 72 + 5 (2)\\f (6) = 72 + 10\\f (6) = 82[/tex]
So, [tex]f (6) = 82[/tex]
ANswer:
[tex]f (6) = 82[/tex]
claire thinks that if she draws a parallelogram with 2 congruent sides, it must be a rhombus. Jacob that she would need to draw a parallelogram with at least 3 congruent sides before she could be sure it was a rhombus. who is correct, if anyone, and why?
Answer:
See below.
Step-by-step explanation:
Neither are correct: a parallelogram with 3 congruent sides could be a square. Also a parallelogram will have 2 pairs of congruent sides but it does not have to be a rhombus.
Answer with explanation:
A rhombus has all sides equal that is four congruent sides.
Statement of Claire
If she draws a parallelogram with 2 congruent sides, it must be a rhombus.
As Rhombus has four Congruent sides.2 Congruent sides does not Guarantee that the Parallelogram will be Rhombus.If Adjacent sides are congruent than Claire statement will be right otherwise not.
Statement of Jacob
If Jacob draws a parallelogram with at least 3 congruent sides ,it must be a rhombus.
Yes ,three congruent sides Guarantees that Parallelogram will be a rhombus because in a Parallelogram opposite sides are equal to each other.So congruency of three sides in a Parallelogram guarantees that this Parallelogram will be a rhombus.
So, we can Conclude that Jacob is correct.
Geometry help needed, please :)
Answer:
C
Step-by-step explanation:
The Hypotenuse is y
The side opposite the given angle (60o) is 12.
You must use one of the trig functions to relate the angle, the side opposite and the hypotenuse.
It turns out that the function you need to use is the sine.
angle = 60o
Side opposite = 12 cm
hypotenuse = h = ???
Sin(60o) = opposite / hypotenuse multiply both sides by the hypotenuse.
hypotenuse * sin(60o) = side opposite
Divide by sin(60o)
hypotenuse = side opposite / sin(60)
hypotenuse = 12/sin(60)
Sin(60) radical form = sqrt(3)/2
hypotenuse = 12 // sqrt(3)/2
hypotenuse = 24 // sqrt(3) Rationalize the denominator.
hypotenuse = 24 * sqrt(3) // ( (sqrt(3)*sqrt(3) )
hypotenuse = 8 sqrt(3)
C
HELP PLEASE!
Kim uses the Fermi process to estimate the number of buckets of rocks she could store in a warehouse. The buckets are shaped like cylinders. The warehouse is shaped like a rectangular prism.
• She estimates the buckets have a height of 25 inches and a diameter of 10 inches.
• She estimates the warehouse is 100 feet long, 50 feet wide, and 20 feet long.
Which expression should Kim use in the process?
Answer:
2×1082×103
Step-by-step explanation:
Just took the unit test. Image as proof
Answer:
The correct option is 3.
Step-by-step explanation:
It is given that the buckets have a height of 25 inches and a diameter of 10 inches.
The volume of a cylinder is
[tex]V=\pi r^2h[/tex]
[tex]V_1=\pi (\frac{10}{2})^2(25)[/tex]
[tex]V_1=\pi (5)^2(25)[/tex]
[tex]V_1=625\pi[/tex]
[tex]V_1=1963.49540849[/tex]
The scientific notation is
[tex]V_1=1.963\times 10^3[/tex]
[tex]V_1\approx 2\times 10^3[/tex]
The warehouse is 100 feet long, 50 feet wide, and 20 feet long.
1 feet = 12 inches
The volume of a cube is
[tex]V=length\times breadth \times height[/tex]
Using the above conversion the volume of cube in cubic inches is
[tex]V_2=(100\times 12)\times (50\times 12)\times (20\times 12)[/tex]
[tex]V_2=172800000[/tex]
The scientific notation is
[tex]V_2=1.728\times 10^8[/tex]
[tex]V_2\approx 2\times 10^8[/tex]
The number of buckets of rocks she could store in a warehouse is
[tex]n=\frac{V_2}{V_1}[/tex]
[tex]n=\frac{2\times 10^8}{2\times 10^3}[/tex]
Therefore the correct option is 3.
The admission fee at fair is $1.50 for children and $4 for adults. On a certain day 2200 people enter the fair and $5050 is collected. Write the system of equations for this scenario if a is the number of adults and c is the number of children
1.50c +4A =2200$ would be the correct awnser
Two sporting goods stores are having discount sales on basketballs at one store a basketball is on sale for 25% off the regular price of 24.95 at the other store the same kind of basketball is on sale for 30% off the regular price of 25.80 what is the difference between the sale prices of the two stores
Answer:
0.66
Step-by-step explanation:
24.95 x 25%=6.23
24.94 - 6.23 = 18.72
25.80 x 30% = 7.74
25.80 - 7.74 = 18.06
18.72 -18 .06 = 0.66
For this case we have:
Store 1:
We propose the following rule of three:
24.95 -----------> 100%
x -------------------> 25%
Where the variable x represents the discount amount:
[tex]x = \frac {25 * 24.95} {100}\\x = 6.2375[/tex]
Thus, the price of the ball is:
[tex]24.95-6.2375 = 18.7125[/tex]
Store 2:
We propose the following rule of three:
25.80 -----------> 100%
x -------------------> 30%
Where the variable x represents the discount amount:
[tex]x = \frac {30 * 25.80} {100}\\x = 7.74[/tex]
Thus, the price of the ball is:
25.80-7.74 = 18.06
Thus, the price difference is:
[tex]18.7125-18.06 = 0.6525[/tex]
ANswer:
0.6525
Am a 5 digit number My tens digit is the sum of my hundreds digit and thousands digit. My tens digit is twice my one digit. My hundreds digit is 1 more than zero. My thousands digit is seven times my hundreds digit. My ten thousands digit is the same as my ones digit.
Answer:
47184
Step-by-step explanation:
1+0=1
7*1=7
7+1=8
4*2=8
A place from this table is chosen at random. Let event A = The place is a city.
What is P(A
c
)?
Answer:
[tex]P(A^c)=\frac{3}{7}[/tex]
Step-by-step explanation:
We have been given a table containing a list of few places that are either city or in North America.
Total number of places in that list = 7
That means sample space has 7 possible events.
Given that a place from this table is chosen at random. Let event A = The place is a city.
Now we need to find about what is [tex]P(A^c)[/tex].
That means find find the probability that chosen place is not a city.
there are 3 places in the list which are not city.
Hence favorable number of events = 3
Then required probability is given by favorable/total events.
[tex]P(A^c)=\frac{3}{7}[/tex]
Answer:
the answer is 3/7
Step-by-step explanation:
PLEASE HELP!!!!!!!!!!!
We know three things:
Angle1 = 90 degrees
Angle2 = 28 degrees
Side = 350 ft
So this means, we are dealing with an AAS triangle (angle, angle, side).
To find side x:
X / sin(90degrees) = 350ft / sin(28degrees)
X = (350ft • sin(90degrees)) / sin(28degrees)
X = 745.5...
=746ft
Makes sense? This is how you solve a AAS triangle. I’m not sure if it’s 100% but, good luck!
Find the volume of the prism.
A. 168 units^3
B. 210 units^2
C. 210 units^3
D. 168 units^2
ANSWER
A. 168 units^3
EXPLANATION
The volume of a pyramid is given by the formula:
[tex]Volume= base \: area \: \times \: vertical \: height[/tex]
Area of triangular base
[tex] = \frac{1}{2} bh[/tex]
[tex] = \frac{1}{2} \times 8 \times 6[/tex]
[tex] = 24 \: {units}^{2} [/tex]
The height of the prism is 7 units.
The volume
[tex] = 24 \times 7[/tex]
[tex] = 168 {units}^{3} [/tex]
Answer:
168
Step-by-step explanation:
Please please help me
Answer:
x = 6.6 cm
Step-by-step explanation:
The angle formed on the circle by the radius and the tangent is right.
Using Pythagoras' identity in the right triangle.
The square on the hypotenuse is equal to the sum of the squares on the other 2 sides.
Hypotenuse is (x + 8.45) and 2 sides are x and 13.5, thus
(x + 8.45)² = x² + 13.5² ← expand left side
x² + 16.9x + 71.4025 = x² + 182.25 ( subtract x² from both sides )
16.9x + 71.4025 = 182.25 ( subtract 71.4025 from both sides )
16.9x = 110.8475 ( divide both sides by 16.9 )
x ≈ 6.6 cm
Help! Help! Help!
What is the distance between the vertices of the graphs corresponding to y = (x - 3)2 and y = x2 - 4?
A.0
B.1
C.5
D.13
Answer:
C.5
Step-by-step explanation:
y = (x - 3)^2
This is in the form y= a(x-h)^2 +k where (h,k) is the vertex
The vertex is at (3,0)
y = x^2 - 4
This is in the form y= a(x-h)^2 +k where (h,k) is the vertex
The vertex is at (0,-4)
The distance between the points is found by
d = sqrt( (x2-x1)^2 + (y2-y2)^2)
= sqrt(( 0-3)^2 + (-4-0)^2)
= sqrt( 9 + 16)
= sqrt(25)
= 5
If two times a certain number is added to 11, the result is 20.
Which of the following equations could be used to solve the problem?
2x = 20
2(x + 11) = 20
2x = 11 + 20
2x + 11 = 20
Answer:
2x + 11 = 20
Step-by-step explanation:
let's start by assigning the variable x to certain number , if we two times a certain number we got 2x, and if we added to 11 the result is 20. So, ordering the equation we will obtain:
2x + 11 = 20
ANSWER
2x+11=20
EXPLANATION
Let the number be x.
Two times this number is 2x
If 11 is added to two times the number, the expression becomes;
2x+11
If the result is 20, then we equate the expression to 20 to get:
2x+11=20
The correct choice is the last option;
A triangle is acute provided all the angles have a measure of less than 90 degrees.
A.Conjunction
Step-by-step explanation:
P(A) = 0.40 P(B) = 0.50 P(A and B) = 0.10 What is P(B/A)
Answer:
Final answer is [tex]P(B|A) = 0.25[/tex].
Step-by-step explanation:
We have been given values of
P(A) = 0.40, P(B) = 0.50, and P(A and B) = 0.10
Now we need to find about what is the value of P(B/A).
Apply formula [tex]P (A \, and \, B)=P(A) \times P(B|A)[/tex]
Plug the given values into above formula:
[tex]P (A \, and \, B)=P(A) \times P(B|A)[/tex]
[tex]0.10 =0.40 \times P(B|A)[/tex]
[tex]\frac{0.10}{0.40} =P(B|A)[/tex]
[tex]0.25 =P(B|A)[/tex]
[tex]P(B|A) = 0.25[/tex]
Hence final answer is [tex]P(B|A) = 0.25[/tex].
What is 102 x 345 =
Answer: 35,190
Step-by-step explanation: If you multiply these together your product will be 35,190
Based on the following set of data, which of the statements shown is true?
11, 11, 12, 13
A.) mean < median
B.) mean = median
C.) mean > median
C.) mean > median
Hope this helps chu
From the following set of data, statement C is true. Option C is correct.
How do you find the data's mean and median?The mean is the proportion of the total number of observations to the sum of the observations.
The median is a number for an organized data set (in ascending or descending order) that has the same number of observations on both sides.
11, 11, 12, 13
The mean of the data set is found as;
mean = (11+11+12+13)/4
mean=47/4
mean=11.75
The median of the data set is;
Arrange the numbers in the ascending order;
11, 11, 12, 13
median= (11+12)/2
median=11.5
mean > median
From the following set of data, statement C is true.
Hence, option C is correct.
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F⃗ (x,y)=−yi⃗ +xj⃗ f→(x,y)=−yi→+xj→ and cc is the line segment from point p=(5,0)p=(5,0) to q=(0,2)q=(0,2). (a) find a vector parametric equation r⃗ (t)r→(t) for the line segment cc so that points pp and qq correspond to t=0t=0 and t=1t=1, respectively. r⃗ (t)=r→(t)= (b) using the parametrization in part (a), the line integral of f⃗ f→ along cc is ∫cf⃗ ⋅dr⃗ =∫baf⃗ (r⃗ (t))⋅r⃗ ′(t)dt=∫ba∫cf→⋅dr→=∫abf→(r→(t))⋅r→′(t)dt=∫ab dtdt with limits of integration a=a= and b=b= (c) evaluate the line integral in part (b). (d) what is the line integral of f⃗ f→ around the clockwise-oriented triangle with corners at the origin, pp, and qq? hint: sketch the vector field and the triangle.
a. Parameterize [tex]C[/tex] by
[tex]\vec r(t)=(1-t)(5\,\vec\imath)+t(2\,\vec\jmath)=(5-5t)\,\vec\imath+2t\,\vec\jmath[/tex]
with [tex]0\le t\le1[/tex].
b/c. The line integral of [tex]\vec F(x,y)=-y\,\vec\imath+x\,\vec\jmath[/tex] over [tex]C[/tex] is
[tex]\displaystyle\int_C\vec F(x,y)\cdot\mathrm d\vec r=\int_0^1\vec F(x(t),y(t))\cdot\frac{\mathrm d\vec r(t)}{\mathrm dt}\,\mathrm dt[/tex]
[tex]=\displaystyle\int_0^1(-2t\,\vec\imath+(5-5t)\,\vec\jmath)\cdot(-5\,\vec\imath+2\,\vec\jmath)\,\mathrm dt[/tex]
[tex]=\displaystyle\int_0^1(10t+(10-10t))\,\mathrm dt[/tex]
[tex]=\displaystyle10\int_0^1\mathrm dt=\boxed{10}[/tex]
d. Notice that we can write the line integral as
[tex]\displaystyle\int_C\vecF\cdot\mathrm d\vec r=\int_C(-y\,\mathrm dx+x\,\mathrm dy)[/tex]
By Green's theorem, the line integral is equivalent to
[tex]\displaystyle\iint_D\left(\frac{\partial x}{\partial x}-\frac{\partial(-y)}{\partial y}\right)\,\mathrm dx\,\mathrm dy=2\iint_D\mathrm dx\,\mathrm dy[/tex]
where [tex]D[/tex] is the triangle bounded by [tex]C[/tex], and this integral is simply twice the area of [tex]D[/tex]. [tex]D[/tex] is a right triangle with legs 2 and 5, so its area is 5 and the integral's value is 10.
A vector parametric equation was computed for the line segment between the points P and Q. This was used to compute the line integral of the vector field along the segment, which was found by substituting the parametrization into the field, deriving it and computing the dot product and integral. The clockwise integral around the triangle was then computed as the negative of the one we obtained earlier.
Explanation:To solve this, consider the given points as vectors, i.e., ℒ = <5,0> and № = <0,2>. First, let's find a vector parametric equation ℝ(τ) for the line segment cc. To do this, we're going to create a vector equation that shows the progression from point ℒ to № with 0 ≤ τ ≤ 1.
Since t changes from 0 to 1, an equation for that movement is ℝ(τ) = (1-τ) * ℒ + τ * №. With insertions, the equation converts to ℝ(τ) = (1-τ) * <5,0> + τ<0,2> = <5-5t,2t>.
For the line integral Along C, our F(x, y) = -yi + xj, when we substitute our ℝ(τ) into this equation, our F becomes F(ℝ) = -2ti + (5-5t)j.
Derivation of ℝ(τ) will give us the rate of change of our function and help in the computation of the line integral. So, ℝ'(τ) = -5i + 2j. The integrand function for line integral then becomes F(ℝ) . ℝ'(τ) = -10t + 10t - 10t2. So our line integral ∫F . dℝ from 0 to 1 will be ∫ (-10t + 10t - 10t2) dt.
For the clockwise integral around the triangle, since our original line integral was done in a counterclockwise direction, it will simply be the negative of the one we just computed.
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50 Points who can actually do this..
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Answer:
x f(x)=1.5^x Function(x,f(x)) Inverse(f(x), x)
0 (1.5)^0=1 (0, 1) (1, 0)
1 b a c
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2 f h i
4 o k d
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Answer:
Functions and x abcd
Step-by-step explanation:
The scale of quantities should always start at zero. True False
Answer:
The correct answer option is true.
Step-by-step explanation:
We are given the following statement and we are to tell if its true or not:
'The scale of quantities should always start at zero'.
To get the correct and accurate measure of any quantity, the scale of that very quantity should always start from zero.
If it does not start from zero, there can be errors in the measure. Therefore, the given statement is true.