The formula for the area of a triangle is A is;
The diameter of a cylinder is 4 m. If the height is triple the radius, which is the closest to the volume of the cylinder?Group of answer choices100.53 space m cubed613.19 space m cubed251.33 space m cubed75.40 space m cubed
Answer:
75.40 cubic meters
Explanation:
Given:
• The diameter of a cylinder is 4 m.
,• The height is triple the radius
We want to find the volume of the cylinder.
First, determine the radius of the cylinder.
[tex]Radius=\frac{Diameter}{2}=\frac{4}{2}=2\;m[/tex]Next, we find the height.
[tex]Height=3\times Radius=3\times2=6\;m[/tex]Substitute these values into the formula for the volume of a cylinder:
[tex]\begin{gathered} V=\pi r^2h \\ =\pi\times2^2\times6 \\ =24\pi \\ =24\times3.14 \\ =75.36 \\ \approx75.40\;m^3 \end{gathered}[/tex]The value closest to the volume of the cylinder is 75.40 cubic meters.
Charlotte has 41 m of fencing to build a three-sided fence around a rectangular plot of land that sits on a riverbank. (The fourth side of the enclosure would be the river.) The area of the land is 204 square meters. List each set of possible dimensions (length and width) of the field.
Answer:
L x W = 8.5 x 24 or 12 x 17
Step-by-step explanation:
x (41 -2x) =204
-2x^2 + 41x -204 = 0 Use Quadratic Formula to find x = 8.5 or 12
(x-8.5)(x-12) = 0
two sides = 8.5 and river/third side = 24
or two sides 12 and river/thirdside = 17
a) 7C₂ =(Simplify your answer.)
This is a combination of the form:
[tex]\begin{gathered} C(n,k)=nCk=\frac{n!}{k!(n-k)!} \\ where: \\ n>k \end{gathered}[/tex]So:
[tex]7C_2=\frac{7!}{2!(7-2)!}=\frac{7!}{2!\cdot5!}=\frac{5040}{2\cdot120}=\frac{5040}{240}=21[/tex]Answer:
21
Molly buys 2 dresses and 3 pairs of pants for $675. The cost of a pair of pants is $25 less than the cost of a dress.a. If a dress cost $x write down, in terms of x, the cost of a pair of pants.b. Form an equation in x and solve itHow much is the cost of i. A dress? ii. A pair of pants
A)
If the dress costs x and the pair of pants cost $25 dollars less, the cost of the pants will be:
[tex]x\text{ - 25}[/tex]B)
We know that the dress costs x and the pants cost x-25. Molly bought 2 dresses and 3 pairs of pants, with the total amount of $675. Thus:
[tex]2x\text{ + 3(x-25) = 675}[/tex][tex]2x\text{ + 3x - 75 = 675}[/tex][tex]5x\text{= 675}+75[/tex][tex]x=\text{ 150}[/tex]If x is the dress cost, so the dress will cost $150. And the pants:
[tex]x\text{ - 25}[/tex][tex]150\text{ - 25}[/tex][tex]125[/tex]The pants will cost $125.
The area of a triangle is 2. Two of the side lengths are 2.1 and 3.8 and the includedangle is acute. Find the measure of the included angle, to the nearest tenth of adegree.
ANSWER
[tex]C=30.1\degree[/tex]EXPLANATION
The area of a triangle using two given sides and the included angle is given as:
[tex]A=\frac{1}{2}a\cdot b\cdot\sin C[/tex]where a and b are the sides and C is the included angle.
Therefore, we have to find C given that:
[tex]\begin{gathered} A=2 \\ a=2.1 \\ b=3.8 \end{gathered}[/tex]Therefore, we have that:
[tex]\begin{gathered} 2=\frac{1}{2}\cdot2.1\cdot3.8\cdot\sin C \\ \Rightarrow\sin C=\frac{2\cdot2}{2.1\cdot3.8}=0.5012 \\ \Rightarrow C=\sin ^{-1}(0.5013) \\ C=30.1\degree \end{gathered}[/tex]That is the measure of the included angle.
in math class, you are checking how a friend balanced an equation. what error did your friend make? explain. unbalanced equation: 16 ÷ 8 =16 ÷ 8 - 1balanced equation: 16 ÷ 8 + 1 = 1 ÷ 8 + 1
Consider the given unbalanced equation which says " 16 divided by 8 " in the Left Hand Side, while it says " 16 divided by 8 then minus one ". This can be represented as,
[tex]\frac{16}{8}=\frac{16}{8}-1[/tex]Consider two things. First, acording to BODMAS rule, division is dprferred over subtraction. Second,
question is in image
The expresion for the correlation coefficient is :
[tex]r=\frac{n\Sigma xy-\Sigma x\Sigma y}{\sqrt[]{\mleft\lbrace n\Sigma x^2-(\Sigma x)^2\}\mleft\lbrace n\Sigma y^2-(\Sigma y)\mright?^2\mright\rbrace}}[/tex]summation of x = 5 + 7 + 10 + 15 + 19
Summation of x = 56
Summation of y = 19 + 17 + 16 + 12 + 7
Summation of y = 71
Summation of prodcut xy
[tex]\begin{gathered} \Sigma xy=5\times19+7\times17+10\times16+15\times12+19\times7 \\ \Sigma xy=687 \end{gathered}[/tex]Summation of x^2 = 25 + 49 + 100 + 225 + 361
Summation of x^2 = 760
Summation of y^2 = 361 + 289 + 256 + 144 + 49
Summation of y^2 = 1099
Substitute tha value in the expression of correlation coefficient
[tex]\begin{gathered} r=\frac{n\Sigma xy-\Sigma x\Sigma y}{\sqrt[]{\mleft\lbrace n\Sigma x^2-(\Sigma x)^2\}\mleft\lbrace n\Sigma y^2-(\Sigma y)\mright?^2\mright\rbrace}} \\ r=\frac{5(687)-56\times71}{\sqrt[]{\mleft\lbrace5(760)-(56)^2\}\mleft\lbrace5(1099\mright)-(71\mright)^2}} \\ r=\frac{541}{\sqrt[]{\begin{cases}3800-3136\}\mleft\lbrace5495-5042\mright\rbrace\end{cases}}} \\ r=\frac{541}{\sqrt[]{300792}} \\ r=\frac{541}{548.44} \\ r=0.985 \end{gathered}[/tex]Answer: A) Correlation coefficient is 0.985
stionsUse the graph at the left to answer the questions in thetable.Your AnswerQuestionIdentify the x-intercept. Usean ordered pair to answer thequestionIdentity the y-intercept. Uzean ordered pair to answer thequestionWhat is the rate or slope ofthe graph?10What is the domain of thegraph?What is the range of thegraph?
Answer:
x-intercept: (-0.5,0)
y-intercept: (0,1)
Slope: 2
Domain: All real values
Range: All real values
Step-by-step explanation:
x-intercept:
The x-intercept is the value of x when y = 0. In this graphic, we have that when y = 0, x = -0.5. So the x-intercept is given by: (-0.5,0)
y-intercept:
The y-intercept is the value of y when x = 0. In this graphic, we have that when x = 0, y = 1. So the y-intercept is given by: (0,1).
Slope:
To find the slope, we select two points. The slope is given by the division of the change in y by the change in x.
Two points: (-0.5, 0) and (0,1)
Change in y: 1 - 0 = 1
Change in x: 0 - (-0.5) = 0 + 0.5 = 0.5
Slope: 1/0.5 = 2
Domain and range:
In a line, the domain is all real values, the same for the range.
Which values are solutions to the inequality below? √x≤=12 check all that apply: A. 143 B. 125 C. 20736 D. 144 E. 12 F. 145
Given data:
[tex]\sqrt[]{x}\leq12[/tex]First squaring both sides we get,
[tex]x\leq144[/tex]Therefore, the value for the solution which satisfies the above equation are
[tex]\begin{gathered} a)\text{ 143}<144 \\ b)\text{ 125}<144 \\ d)\text{ 144}\leq144 \\ e)\text{ 12}<144 \end{gathered}[/tex]Thus, the ans is (a) , (b) , (d) , (e)
In a recent awards ceremony, the age of the winner for best actor was 38 and the age of the winner for best actress was 51. For all best actors, the mean age is 46.8 years and the standard deviation is 5.7 years. For all best actresses, the mean age is 31.9 years and the standard deviation is 10.8 years. (All ages are determined at the time of the awards ceremony.) Relative to their genders, who had the more extreme age when winning the award, the actor or the actress? Explain.
Given
the age of the winner for best actor was 38 and the age of the winner for best actress was 51.
For all best actors, the mean age is 46.8 years and the standard deviation is 5.7 years.
For all best actresses, the mean age is 31.9 years and the standard deviation is 10.8 years.
Find
who had the more extreme age when winning the award, the actor or the actress
Explanation
the Z- score is given by
[tex]Z=\frac{X-\mu}{\sigma}[/tex]Best Actor
X = 38
mean = 46.8 years
standard deviation = 5.7 years.
so ,
[tex]\begin{gathered} Z=\frac{38-46.8}{5.7} \\ \\ Z=-1.54385964912\approx-1.54 \end{gathered}[/tex]Best Actress
X = 51
mean = 31.9 years
standard deviation = 10.8 years.
so ,
[tex]\begin{gathered} Z=\frac{51-31.9}{10.8} \\ \\ Z=1.76851851852\approx1.77 \end{gathered}[/tex]the best actress's age is farther from the mean so he has the more extreme age when the winning the award.
Final Answer
Hence , the z- score for best actor is -1.54 and for best actress = 1.77
Actress has more extreme age
0,2% of what number is 8?
Answer: The number is 4,000.
Step-by-step explanation:
First, we will turn 0.2% into a decimal.
0.2% / 100 = 0.002
Next, we will set up an equation. Let x be equal to "what number:"
0.2% of what number is 8 ➜ 0.002x = 8
Lastly, we will solve by dividing both sides by 0.02.
0.002x = 8
x = 4,000
If we wish to, we can check our answer.
4,000 * 0.002 = 8
Read more about this question here: https://brainly.com/question/14091059
Suppose the department of motor vehicles in a state uses only six spaces and the digits 0 to 9 create it's license plates. Digits can be repeated
Using the Fundamental Counting Theorem, the number of possible plates is given as follows:
1,000,000.
Fundamental Counting TheoremThe Fundamental Counting Theorem states that if there are n independent trials, each with [tex]n_1, n_2, \cdots, n_n[/tex] possible results, the total number of outcomes is given according to the following rule:
[tex]N = n_1 \times n_2 \times \cdots \times n_n[/tex]
In this question, the plate is composed by six independent digits, hence the parameters are given as follows:
[tex]n_1 = n_2 = n_3 = n_4 = n_5 = n_6 = 10[/tex]
(as there are 10 possible digits, from 0 to 9).
Hence the number of possible plates is calculated as follows:
N = 10 x 10 x 10 x 10 x 10 x 10 x 10 = 10^6 = 1,000,000.
Missing information
This problem is incomplete, hence we suppose that it asks for how many plates can be built.
More can be learned about the Fundamental Counting Theorem at https://brainly.com/question/24314866
#SPJ1
A group fitness gym classifies its fitness class attendees by class type and member status. The marketing team has gathered data from a random month, in which therewere 2153 class attendees. The data is summarized in the table below.Class Type and Member Status of ClassAttendeesClass TypeBarreBoot CampBoxingSpinningYogaMember207230291233239Non-Member196185180176216What is the probability that an attendee does not attend a barre class? Enter a fraction or round your answer to 4 decimal places, if necessary.
The total number associated to barre class is 207 + 196 = 403
Therefore, the probability that a attendee does not attend a barre class is given by 403/2153 = 0.1872
Probability of heads and tails
SOLUTION
A coin was flipped.
A = probability of it landing on a head
B = probability of it landing on a tail.
We are asked P(A and B), that means probability of landing on a head and on a tail.
A coin can only land on a head or it can land on a tail. It can't land on a head and a tail at the same time.
Therefore the probability of landing on a head and a tail is 0.
Hence, the answer is 0.
4.5(8-x) -36 = 102 - 25 (3x + 24)
Let's simplify the following equation
Chandrasekhar Subramaayan is taking a course in Astrophysics at the University of Madras. Tomorrow, he will be taking the Final Exam. The Final Exam makes up 65% of the overall average. Chandrasekhar Subramanyan's current average is 74%. What is the lowest score that he can get on the Final Exam that would increase his overall average to a “B”? Show your work.
Final Exam accounts for 65% of total.
The rest is 100 - 65 = 35%
On the 35%, he has 74% or 74 out of 100.
Let's say the final exam is out of 100 and we want to find how much he can afford to get to take him to "B".
Let's say his score needs to be "x" in order to get B.
The overall weighted average of the course would be counted as:
[tex]35\%of74\%+65\%of\frac{x}{100}=83\%oftotal[/tex]Breaking it down:
74 out of 100 means WHAT out of 35? (let's call the unknown "y")
[tex]\frac{74}{100}=\frac{y}{35}[/tex]y would be:
[tex]\begin{gathered} \frac{74}{100}=\frac{x}{35} \\ 100x=74\cdot35 \\ x=25.9 \end{gathered}[/tex]So, he currently has 25.9 out of 35.
He would need 83 to get overall average to 83%.
He would need:
83 - 25.9 = 57.1 out of 65
That is:
[tex]\frac{57.1}{65}\cdot100=87.85\%[/tex]Basically he would need a percentage of 87.85% (MINIMUM) on the final exam to make his overall average to 83% (which is the bare minimum for a B grade).
What property is used in theses math problems
1: If AB=CD, then CD=AB
2: 5(x-4) = 5x-20
3: 3x-17=8, then 3x=25
4: 22m=440, then m=20
5:(2a+3a)-4a=7a+55, then (5a)-4a=7a+55
6:If GH=2x+17, and 2x+17=JK, then GH=JK
7:2b+(b-7)=17, then (2b+b)-7=17
8:(x+7)/3=13, then x+7=39.
9:m
10:4+x=13, then x+4=13
The various algebraic properties used for each of the given expressions are;
1) Symmetric property of equality.
2) Distributive property of equality.
3) Addition property of equality.
4) Division property of equality.
5) Addition property of equality.
6) Transitive property of equality.
7) Associative property of equality.
8) Multiplication property of equality
9) Commutative property of equality.
What is the Property of Algebra that is used?
There are different properties of equality or algebra such as;
Commutative PropertyAssociative PropertyDistributive PropertyInverse propertyTransitive PropertyReflexive PropertySymmetric propertySubstitution propertyAddition propertySubtraction property1) If AB=CD, then CD=AB; Property used here is Symmetric property of equality because it suits the definition.
2) 5(x-4) = 5x-20; Property used here is Distributive property of equality because it suits the definition.
3) 3x - 17 = 8, then 3x = 25; Property used here is Addition property of equality because it suits the definition.
4) 22m=440, then m=20; Property used here is Division property of equality because it suits the definition.
5) :(2a+3a)-4a=7a+55, then (5a)-4a=7a+55; Property used here is Addition property of equality because it suits the definition.
6) If GH=2x+17, and 2x+17=JK, then GH=JK; Property used here is Transitive property of equality because it suits the definition.
7) 2b+(b-7)=17, then (2b+b)-7=17; Property used here is Associative property of equality because it suits the definition.
8) (x+7)/3=13, then x+7=39.; Property used here is Multiplication property of equality because it suits the definition.
9) m 10:4+x=13, then x+4=13; Property used here is Commutative property of equality because it suits the definition.
Read more about Properties of Algebra at; https://brainly.com/question/2284285
#SPJ1
How many mL of a 0.6% solution can be made with 6mg of a drug? Round your final answer to 1 decimal place if necessary.
ANSWER :
EXPLANATION :
You wish to program an LED strip containing n LEDs . Each individual LED can be lit in red, green, blue, or white. How many length-n
colour sequences have at least one pair of adjacent LEDs that light up with the same colour?
Hint: Find a recurrence relation. Order matters, (GGR and RGG are different). Also need to consider for the intersection of probabilities.
Using the Fundamental Counting Theorem, the number of length-n sequences that have at least one pair of adjacent LEDs that light up with the same color is:
[tex]N = 4 - 4 \times 3^{n - 1}[/tex]
What is the Fundamental Counting Theorem?It is a theorem that states that if there are n trials, each with [tex]n_1, n_2, \cdots, n_n[/tex] possible results, each thing independent of the other, the number of results is given as follows:
[tex]N = n_1 \times n_2 \times \cdots \times n_n[/tex]
In the context of this problem, there are n LEDs, each with 4 possible colors, hence he total number of sequences is:
[tex]T = 4^n[/tex]
For sequences with no repeating LEDs, we have that:
The first LED can have 4 outcomes.Any of the following LEDs can have 3 outcomes. (except the color of the previous one).Hence the number is:
[tex]T_n = 4 \times 3^{n-1}[/tex]
(n - 1) as the first LED is not included on those with 3 possibilities.
Having at least one repeating and no repeating are complementary events, hence the number of sequences is:
[tex]N = 4 - 4 \times 3^{n - 1}[/tex]
More can be learned about the Fundamental Counting Theorem at https://brainly.com/question/24314866
#SPJ1
5. If an integer is randomly selected from all positive 2-digit integers, what is the probability
that the integer chosen has a 4 in the tens place?
(a) 1/6
(b) 1/8
(c) 1/4
(d) 1/9
(e) 2/9
A probability is a numerical representation of the likelihood or chance that a specific event will take place. The value of P(A) = 1/9.
What is meant by probability?Probability is a branch of mathematics that deals with numerical representations of the likelihood of an event occurring or of a proposition being true. The probability of an event is a number between 0 and 1, with 0 approximately denoting impossibility and 1 denoting certainty.
A probability is a numerical representation of the likelihood or chance that a specific event will take place. Both proportions ranging from 0 to 1 and percentages ranging from 0% to 100% can be used to describe probabilities.
Total number of 2-digit terms =(99 - 10) + 1 = 90
n(s) = 90
Number of 2 digits that have 4 in tens place = 40, 41, 42 ......49
n(a) = 10
P(A) = n(a)/n(s)
Substituting the values in the above equation, we get
P(A) = 10/90
P(A) = 1/9
The value of P(A) = 1/9.
Therefore, the correct answer is option (d) 1/9.
To learn more about probability refer to:
https://brainly.com/question/25870256
#SPJ13
a line perpendicular to y = 2x +4 through (4.2) slope of the new line equationequation of the new line
y= 2x+4
Slope-intercept form
y=mx+b
Where:
m= slope
b= y-intercept
y=2x+4
Slope = m= 2
Perpendicular lines have negative reciprocal slopes.
So, the slope of the new line = -1/2
New equation:
y= -1/2x+b
Replace x,y by the point (4,2) and solve for b:
2 = -1/2(4)+b
2= -2 + b
2+2 = b
4 = b
New line equation:
y= -1/2x+4
50 points !!!!!!!!!!!!!
PLUG IN 2 IN THE PLACE OF X IN THE FUNCTION THEN SIMPLIFY
f(2)= |5×2|
f(2)=|10|
[tex]f(2) = 10[/tex]
ATTACHED IS THE SOLUTION
Answer:
[tex]\rm f(2)=|5\;^*\;2|=|10|=10[/tex]
Step-by-step explanation:
The bars either side of an expression or a value are the absolute value symbol. "Absolute value" means how far a value is from zero. Therefore, the absolute value of a number is its positive numerical value.
Given absolute value function:
[tex]f(x)=|5x|[/tex]
To find f(2), substitute x = 2 into the given function:
[tex]\begin{aligned}\implies f(2)&=|5 \cdot 2|\\&=|10|\\&=10\end{aligned}[/tex]
The graph of y=h(x)y=h(x)y, equals, h, left parenthesis, x, right parenthesis is a line segment joining the points (1,9)(1,9)left parenthesis, 1, comma, 9, right parenthesis and (3,2)(3,2)left parenthesis, 3, comma, 2, right parenthesis.
Drag the endpoints of the segment below to graph y=h^{-1}(x)y=h
−1
(x)y, equals, h, start superscript, minus, 1, end superscript, left parenthesis, x, right parenthesis.
The graph of the linear function and it's inverse is given at the end of the answer.
The inverse function is: [tex]h^{-1}(x) = -6x + 55[/tex]
How to find the inverse function?To find the inverse function, we exchange x and y in the original function, then isolate y.
In the context of this problem, first we have to find the function h(x), which is a linear function going through points (1,9) and (3,2), hence the slope is given by:
m = (2 - 1)/(3 - 9) = -1/6.
Then:
y = -x/6 + b.
When x = 1, y = 9, hence we can find the intercept b as follows:
9 = -1/6 + b
b = 54/6 + 1/6
b = 55/6.
Then the equation is:
h(x) = -x/6 + 55/6
Then:
y = -x/6 + 55/6
6y = -x + 55
6x = -y + 55
y = -6x + 55.
Hence the inverse function is:
[tex]h^{-1}(x) = -6x + 55[/tex]
More can be learned about inverse functions at https://brainly.com/question/3831584
#SPJ1
Write an equation in standard form for the line that passes through the given points.
(−4, 9), (2,−9)
Answer:
3x + y = -3
Step-by-step explanation:
(-4, 9), (2, -9)
(x₁, y₁) (x₂, y₂)
y₂ - y₁ -9 - 9 -18 -18
m = ----------- = ----------- = ---------- = ------- = -3
x₂ - x₁ 2 - (-4) 2 + 4 6
y - y₁ = m(x - x₁)
y - 9 = -3(x - (-4)
y - 9 = -3(x + 4)
y - 9 = -3x - 12
+9 +9
---------------------
y = -3x - 3
Standard Form:
-3x + y = -3
I hope this helps!
A dairy needs385 gallons of milk containing5 % butterfat. How many gallons each of milk containing8 % butterfat and milk containing1 % butterfat must be used to obtain the desired385 gallons?
The first step to this problem is finding the amount of butterfat there are in the 385 gallons milk with 5% of butterfat in it. To do that we need to multiply the percentage by the total amount of milk. This is done below:
[tex]\begin{gathered} \text{butterfat }_1=5\text{ \%}\cdot385 \\ \text{butterfat }_1=\frac{5}{100}\cdot385 \\ \text{butterfat }_1=0.05\cdot385=19.25 \end{gathered}[/tex]There are 19.25 gallons of butterfat on the final mixture, the amount of butterfat from each of the milks we are going to mix need to be equal to that. The first milk contains 8% of butterfat, while the second milk contains 1%. So we have:
[tex]\text{butterfat }_2=x\cdot8\text{ \%}\cdot=0.08\cdot x[/tex][tex]\text{butterfat}_3=y\cdot1\text{ \%}=0.01\cdot y[/tex]Where "x" is the number of gallons from the first milk and "y" is the number of gallons from the second milk. The number of gallons of each milk, when added, should be equal to the number of gallons on the final milk, so we have:
[tex]x+y=385[/tex]The same is valid for the amount of butterfat.
[tex]0.08\cdot x+0.01\cdot y=19.25[/tex]We have now created a system of equations as shown below:
[tex]\mleft\{\begin{aligned}x+y=385 \\ 0.08\cdot x+0.01\cdot y=19.25\end{aligned}\mright.[/tex]To solve this system we will multiply the first equation by "-0.01":
[tex]\{\begin{aligned}-0.01x-0.01y=-3.85 \\ 0.08\cdot x+0.01\cdot y=19.25\end{aligned}[/tex]Now we need to add both equations.
[tex]\begin{gathered} -0.01x+0.08x-0.01y+0.01y=-3.85+19.25 \\ 0.07x=15.4 \\ x=\frac{15.4}{0.07}=220 \end{gathered}[/tex]To find the value of "y" we will use the first equation:
[tex]\begin{gathered} 220+y=385 \\ y=385-220=165 \end{gathered}[/tex]We need 220 gallons from the 8% butterfat milk and 165 gallons from the 1% butterfat milk.
The width of a room is 7 feet, and the area of the room is 77 square feet. Find the room's length.
...
Answer:
11
Step-by-step explanation:
A = lw
l = A/w
l = 77/7
l = 11
Hope that helps
For each relation, decide whether or not it is a function.
Top left: Yes, each input corresponds to only one output.
Top right: Yes, each input corresponds to only one output.
Bottom left: Yes, each input corresponds to only one output.
Bottom right: No, the input of -1 corresponds to three different outputs.
Simplify 4 - 2y +(-8y) +7.3.4-2y+ ( 8y) +7.3=Q(Simplify your answer. Use integers or decimals for any numbers in theWhat is 4-2y+(-8y)+7.3
In order to simplify an equation we want to add all the terms with the same unkowns.
In the equation
4 - 2y + (-8y) + 7.3
we can see that there are terms with y and numbers alone.
We combine like terms:
Terms with y: -2y,
Write the equation of the line with slope 1/4 that passes through the point (0, -1).
The point-slope form of a line is given by:
y - k = m ( x - h)
Where m is the slope of the line and (h, k) is a point through which the line passes.
We are given the values of m = 1/4 and the point (0, -1).
Substituting:
[tex]\begin{gathered} y-(-1)=\frac{1}{4}(x-0) \\ \text{Simplifying:} \\ y+1=\frac{1}{4}x \\ \text{Solving for y:} \\ y=\frac{1}{4}x-1 \end{gathered}[/tex][tex]\int\limits^(6,4,8)_(1,0,-3) {x} \, dx + y\, dy - z\, dz[/tex]
Notice that [tex](x,y,-z)[/tex] is a gradient field:
[tex]\nabla f(x,y,z) = (x,y,-z) \implies \begin{cases} f_x = x \\ f_y = y \\ f_z = -z \end{cases}[/tex]
That is, there exists a scalar function [tex]f(x,y,z)[/tex] whose gradient is the given vector field. Solve for [tex]f[/tex].
[tex]\displaystyle \int f_x \, dx = \int x \, dx \implies f(x,y,z) = \frac12 x^2 + g(y,z)[/tex]
[tex]f_y = g_y = y \implies \displaystyle \int g_y \, dy = \int y \, dy \implies g(y,z) = \frac12 y^2 + h(z)[/tex]
[tex]f_z = h_z = -z \implies \displaystyle \int h_z \, dz = - \int z \, dz \implies h(z) = -\frac12 z^2 + C[/tex]
[tex]\implies f(x,y,z) = \dfrac{x^2+y^2-z^2}2 + C[/tex]
By the gradient theorem, it follows that
[tex]\displaystyle \int_{(1,0,-3)}^{(6,4,8)} x \, dx + y \, dy - z \, dz = f(6,4,8) - f(1,0,-3) = \boxed{-2}[/tex]