She spends the cost of $0.46 if she purchases the items in the city where she lives.
What is the percentage?The percentage is defined as a ratio expressed as a fraction of 100.
The given data below as:
Cost of the shirt = $11.99
Cost of the shoes = $35.50
Cost of the belt = $6.75
Cost of the socks = $3.00
Total cost of the items = 11.99 + 35.50 + 6.75 + 3.00
Total cost of the items = $57.24
In the city she lives in, the sales tax rate is 7.15%.
Total cost of the items with tax of 7.15%
= 57.24 × 7.15/100 + 57.24 = 4.092 + 57.24 = $61.33
In another city, the sales tax rate is 6.35%
Total cost of the items with tax of 6.35%
= 57.24 × 6.35/100 + 57.24 = 3.633+ 57.24 = $60.87
Hence, she spends the cost of $0.46 if she purchases the items in the city where she lives.
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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 :
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.
B. f(x) = x² + 1
4. Evaluate f(-3).
5. Evaluate f (6).
6. Circle any ordered pairs
that are included in the
function:
(0, -1) (5, 17) ((-7,50)
Hey! Can anybody help me with this?I don't need a very big explanation just a very brief explanation leading to the answer as I already kinda know this stuff. Thanks!
In an ordered pair (x,y), x denotes the domain of the relation, and y denotes the range of the relation.
Knowing this, the three ordered pairs that will form a relation with a range of {-3, 4, 7} are
(-1, -3), (0, 4), and (2,7).
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]
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Cuanto es 2 1/2 + 1 3/4 + 1/2=?
Ayudaaa
Answer:
2 2/4 + 1 3/4 + 2/4 = 4 3/4
4 3/4
Answer:
4 and 3/4
Step-by-step explanation:
2 1/2
+1 3/4
+ 1/2
_____
4 and 3/4
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]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.
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,
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
subtract 5x from 7x-6
We can subtract 5x from 7x - 6 like this:
7x - 6 - 5x
Now, we just have to combine like terms, in this case, the terms that have the x variable, like this:
7x - 5x - 6
2x - 6
Then, after subtracting 5x from 7x - 6 we get 2x - 6
Picture listed below help.
Answer:
Perimeter = 65.12 inches, Area = 292.48 square inches
Step-by-step explanation:
Perimeter:
Total perimeter = perimeter of semicircle + perimeter of rectangle (note: excluding the shared line, since that is not on the outside of the shape).
= [tex]\frac{1}{2} 2\pi r[/tex]
= [tex]\frac{1}{2} *2*3.14*8[/tex]
= 25.12
Perimeter = 25.12 + 12 + 12 + 16 = 65.12 inches
Area:
Total area = area of semicircle + area of rectangle
Area of semicircle = [tex]\frac{1}{2} \pi r^{2}[/tex]
= [tex]\frac{1}{2} *3.14* 8^{2}[/tex]
= 100.48
Area of rectangle = l * w
= 16 * 12
= 192
Total area = 100.48 + 192 = 292.48
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
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I don't understand this at all. Solve the equation for T1.
Answer:
T1 = P2 * V2 / T2 * P1 * V1
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.
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.
May u please help me with my geometry study guide Only 2 questions
Answer:
4.
[tex]x=AB=\frac{5\sqrt[]{2}}{2}[/tex]5.
[tex]BC=x=5\sqrt[]{3}[/tex]Step-by-step explanation:
Relationships in a right triangle:
The sine of an angle is the length of the opposite side to the angle divided by the hypotenuse.
The cosine of an angle is the length of the adjacent side to the angle divided by the hypotenuse
The tangent of an angle is the length of the opposite side to the angle divided by the length of the adjacent side to the angle.
Question 4:
hypotenuse is 5.
AB is adjacent to an angle of 45º. So
[tex]\sin 45^{\circ}=\frac{x}{5}^{}^{}[/tex][tex]\frac{\sqrt[]{2}}{2}=\frac{x}{5}[/tex]Applying cross multiplication:
[tex]2x=5\sqrt[]{2}[/tex][tex]x=AB=\frac{5\sqrt[]{2}}{2}[/tex]Question 5:
Hypotenuse is 10.
BC is opposite to an angle of 60. So
[tex]\sin 60^{\circ}=\frac{x}{10}[/tex][tex]\frac{\sqrt[]{3}}{2}=\frac{x}{10}[/tex][tex]2x=10\sqrt[]{3}[/tex][tex]x=\frac{10\sqrt[]{3}}{2}[/tex][tex]BC=x=5\sqrt[]{3}[/tex]Least squares regression line is y=4x-8 what is the best predicted value for y given x =7?
We have a regression model to predict y from the value of x.
The equation of this regression line is:
[tex]y=4x-8[/tex]We then can calculate y for x = 7 as:
[tex]y(7)=4(7)-8=28-8=20[/tex]Answer: the predicted value of y when x = 7 is y(7) = 20
A farmer harvested 192 pounds of Yukon Gold potatoes and 138 pounds of russet potatoes. He divided the Yukon
Gold potatoes evenly among 8 baskets. He divided the russet potatoes evenly among 6 baskets. He then filled a
sack with one basket of Yukon Gold potatoes and one basket of russet potatoes.
Which equation could be used to solve for n, the number of number of pounds of potatoes the farmer put in the
sack?
The farmer put a total of 47 potatoes in one sack.
Here, we are given that a farmer harvested 192 pounds of Yukon Gold potatoes and 138 pounds of russet potatoes.
He divided the Yukon Gold potatoes evenly among 8 baskets.
Thus, the number of pounds of potatoes in each basket = 192/8
= 24
Similarly, he divided the russet potatoes evenly among 6 baskets.
Thus, the number of pounds of russet potatoes in each basket = 138/6
= 23
Now, he fills a sack with one basket of Yukon Gold potatoes and one basket of russet potatoes.
n = the number of pounds of potatoes the farmer put in the sack
Thus, n = 24 + 23
n = 47
Thus, the farmer put a total of 47 pounds of potatoes in one sack.
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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 ball is thrown in the air from a platform that is 96 feet above ground level with an initial vertical velocity of 32 feet per second. The height of the ball, in feet, can be represented by the function shown where t is the time, in seconds, since the ball was thrown. Rewrite the function in the form that would be best used to identify the maximum height of the ball and find Approximately when the object lands on the ground when rounded to the nearest tenth.
Answer:
y = -16 (x - 1)^2 + 112
The object lands on the ground in approximately 3.6s
Explanation:
The equation given is that of a parabola.
Now the maximum (local) point of a parabola is the vertex. Therefore, if we want to rewrite our function in the form that would be used to find the maximum height, then that form must be the vertex form of a parabola.
The vertex form of a parabola is
[tex]y=a(t-h)^2+k[/tex]where (h, k) is the vertex.
The only question is, what is the vertex for our function h(t)?
Remember that if we have an equation of the form
[tex]y=ax^2+bx+c[/tex]then the x-coordinate of the vertex is
[tex]h=-\frac{b}{2a}[/tex]Now in our case b = 32 and a = -16; therefore,
[tex]h=\frac{-32}{2(16)}=1[/tex]We've found the value of the x-coordinate of the vertex. What about the y-coordinate? To get the y-coordinate, we put x = 1 into h(t) and get
[tex]k=-16(1)+32(1)+96=112[/tex]Hence, the y-coordindate is k = 112.
Therefore, the vertex of the parabola is (1, 112).
With the coordinates of the vertex in hand, we now write the equation of the parabola in vertex form.
[tex]h(t)=a(t-1)^2+112[/tex]The only problem is that we don't know what the value of a is. How do we find a?
Note that the point (0, 96) lies on the parabola. In other words,
[tex]h(0)=-16(0)^2+32(0)+96=96[/tex]Therefore, the vertex form of the parabola must also contain the point (0, 96).
Putting in t = 0, h = 96 into the vertex form gives
[tex]96=a(0-1)^2+112[/tex][tex]96=a+112[/tex]subtracting 112 from both sides gives
[tex]a=-16[/tex]With the value of a in hand, we can finally write the equation of the parabola on vertex form.
[tex]\boxed{h\mleft(t\mright)=-16\left(t-1\right)^2+112.}[/tex]Now when does the object hit the ground? In other words, for what value of t is h(t) = 0? To find out we just have to solve the following for t.
[tex]h(t)=0.[/tex]We could either use h(t) = -16t^2 + 32t + 96 or the h(t) = -16(t - 1)^2 + 112 for the above equation. But it turns out, the vertex form is more convenient.
Thus we solve,
[tex]-16\left(t-1\right)^2+112=0[/tex]Now subtracting 112 from both sides gives
[tex]-16(t-1)^2=-112[/tex]Dividing both sides by -16 gives
[tex](t-1)^2=\frac{-112}{-16}[/tex][tex](t-1)^2=7[/tex]taking the square root of both sides gives
[tex]t-1=\pm\sqrt{7}[/tex]adding 1 to both sides gives
[tex]t=\pm\sqrt{7}+1[/tex]Hence, the two solutions we get are
[tex]t=\sqrt{7}+1=3.6[/tex][tex]t=-\sqrt{7}+1=-1.6[/tex]Now since time cannot take a negative value, we discard the second solution and say that t = 3.6 is our valid solution.
Therefore, it takes about 3.6 seconds for the object to hit the ground.
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.
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.
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The turning points of the graph are (-1.73, -10.39) and (1.73, 10.39). What is the range of the Polynomial function f?
Given the turning points of the graph:
(-1.73, -10.39) and (1.73, 10.39).
Let's determine the range of the graphed function.
The range of a function is the set of all possible values of y.
The y-values are represented on the vertical axis.
From the graph, we can see the function goes up continuously and goes down continuously.
Therefore, we can say the range of the function is real numbers.
Therefore, the range of the function in interval notation is:
(-∞, ∞)
ANSWER:
rRange: (-∞, ∞)
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.
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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).
−8−x=−3(2x−4)+3x (with dcmam)
x = 10 is the solution to the equation -8 - x = -3( 2x - 4 ) + 3x using DCMAM method.
What is the solution to the given equation?
Given the equation in the question;
-8 - x = -3( 2x - 4 ) + 3xx = ?First, apply the distributive property to eliminate the parenthesis.
-8 - x = -3( 2x - 4 ) + 3x
-8 - x = -3×2x -3×-4 + 3x
Multiply -3 × 2x
-8 - x = -6x -3×-4 + 3x
Multiply -3 × -4
-8 - x = -6x + 12 + 3x
Next, combine like terms.
Add -6x and 3x
-8 - x = -3x + 12
Next, move variable to one side and constant terms to the other.
-x + 3x = 12 + 8
Add -x and 3x
2x = 12 + 8
Add 12 and 8
2x = 20
Divide both sides by 2
2x/2 = 20/2
x = 20/2
x = 10
Therefore, the value of x is 10.
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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]
[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]
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.