[tex]\vec r(t)=14t^4\,\vec\imath+t^6\,\vec\jmath[/tex]
[tex]\mathrm d\vec r=(56t^3\,\vec\imath+6t^5\,\vec\jmath)\,\mathrm dt[/tex]
[tex]\vec f(x,y)=xy\,\vec\imath+6y^2\,\vec\jmath\implies\vec f(x(t),y(t))=14t^{10}\,\vec\imath+6t^{12}\,\vec\jmath[/tex]
Then the line integral is
[tex]\displaystyle\int_C\vec f\cdot\mathrm d\vec r=\int_0^1(14t^{10}\,\vec\imath+6t^{12}\,\vec\jmath)\cdot(56t^3\,\vec\imath+6t^5\,\vec\jmath)\,\mathrm dt[/tex]
[tex]=\displaystyle\int_0^1(36t^{17}+784t^{13})\,\mathrm dt=\boxed{58}[/tex]
The line integral c f dot dr for the given parameters can be calculated by substituting values of functions r(t) and f(x, y) into the line integral, then integrating over t from 0 to 1.
Explanation:To solve this problem, we start by writing the vector valued function r(t) and the vector field f(x, y) in component form. Given that r(t) = 14t4i + t6j and f(x, y) = xyi + 6y2j, the line integral c f dot dr becomes a definite interval where we integrate over t from 0 to 1.
In this case, we're essentially finding the work done by the vector field f as a particle moves along the path described by r(t). To calculate this, we need to find dr/dt which in our case evaluates to 56t3i + 6t5j.
Subsequent substitution of x = 14t4 and y = t6 into f will provide a new vector ft for f dot dr. Now we can find f dot dr by multiplying corresponding components of ft and dr/dt, add them up, and then integrate over t from 0 to 1.
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Find the missing lengths of the sides.
Answer:
b = 8√3 and c = 16
Step-by-step explanation:
Points to remember
If the angles of a right angled triangle with angles are 30°, 60, and 90 the their sides are in the ratio 1 : √3 : 2
To find the unknown side lengths
From the given figure we can see a right angled triangle
Angles are 30°, 60° and 90°
sides are in the ratio 1 : √3 : 2
1 : √3 : 2 = 8 : b : c
b = 8√3 and c = 8 * 2 = 16
Therefore b = 8√3 and c = 16
Franco needs to pack 125 bottles of juice. Each box holds 8 bottles. How many boxes are needed to pack all of the juice?
Answer:
16 Boxes
Step-by-step explanation:
Total Bottles:125
Amount of Bottles A Box Is Able To Hold:8
125/8=15.625
Round Up To Nearest Whole Number
16 Boxes Are Needed To Hold 125 Bottles
Nicole's cleaning company uses a linear model to determine the amount they charge their clients. They charged $144 for a cleaning job that took 4 hours. They charged $188 for a cleaning job that took 6 hours. Which equation could be used to determine the amount charged, C, for a cleaning job that takes t hours?A.C = 10t + 104B.C = 36t C.C = 44t + 56D.C = 22t + 56
Answer:
D
Step-by-step explanation:
Let the following point represent the first job: (4, $144). Let (6, $188) represent the second. Then we find the equation of the line through these two points. As we go from (4, $144) to (6, $188), x increases by 2 and y increases by $44. Thus, the slope of this line is m = rise/run = $44/(2 hrs), or $22/hr.
Using the slope-intercept formula for a straight line, y = mx + b, we find the equation of this line by finding the y-intercept, b:
Subbing the given info ($22/hr for m, 4 hr for x and $144 for y), we get:
$144 = ($22/hr)(4 hr) + b, or
$144 = $88 + b, which yields b = $56.
Thus, the desired equation is y = mx + b, or y = ($22/hr)x + $56. This matches the last answer (D)
Which of the following are true statements? Check all that apply.
Answer:
Your answer is D.
Step-by-step explanation:
Although A and C are also correct, according to a graph using the free online program Desmos
PLEASE HELP ASAP!!! CORRECT ANSWER ONLY PLEASE!!!
The graph of the function f(x) = x^3 – 7x – 6 intersects the x-axis at the points (–2, 0), (–1, 0), and (3, 0) as shown.
Which expression is equivalent to x^3 – 7x – 6?
The answer is D, (x + 1) (x + 2) (x - 3)
Answer: D) (x + 1))x + 2)(x - 3)
Step-by-step explanation:
The x-intercepts help us to form the equation of the curve:
(-2, 0) (-1, 0) and (3, 0)
--> x = -2, x = -1, x = 3
--> x + 2 = 0 x + 1 = 0 x - 3 = 0
--> (x + 2) × (x + 1) × (x - 3) = 0
This is the reverse order for the Zero Product Property
Let f(x)=14/ 7+2e^−0.6x .
What is f(3) ?
Answer:
[tex]f(3)=1.9[/tex]
Step-by-step explanation:
we have
[tex]f(x)=\frac{14}{7+2e^{-0.6x}}[/tex]
we know that
f(3) is the value of the function for the value of x equal to 3
so
substitute the value of x=3 in the function
[tex]f(3)=\frac{14}{7+2e^{-0.6(3)}}=1.9[/tex]
Can someone help me out?
Answer:
40 degrees
Step-by-step explanation:
A triangle solver tool can find the angle easily. It is 39.8°, which rounds to 40°. Apps are available on some calculators, on the Internet, and for iOS and Android phones and tablets.
___
You may be expected to solve this using the Law of Cosines. If we name the sides ...
a = 1.75b = 3.00c = 2.00the law of cosines tells us the relationship is ...
c² = a² + b² -2ab·cos(θ)
Then the angle is ...
θ = arccos((a² +b² -c²)/(2ab)) = arccos((3.0625 +9 -4)/(2·1.75·3))
= arccos(8.0625/10.5) ≈ 39.838° ≈ 40°
Which equation shows how (10, 8) can be used to write the equation of this line in point-slope form?
Answer:
y - 8 = m(x - 10). (h, k)
Step-by-step explanation:
If the slope is m, then the desired equation is y - 8 = m(x - 10). (h, k) represents the given point, which here is (10, 8).
Next time, won't you please share the answer choices?
Answer: y - 8 + 10x
ep-by-step explanation:
PLEASE HELP!!
Which scale factors produce an expansion under a dilation of he original image?
Choose all answers that are correct
a)-2
b)-1/2
c)1/2
d)2
Under the dilation, the point (3,5) is moved to (6,10)
What is the scale factor of the dilation?
For dilation's, you can ignore any negative signs.
An expansion would be a number greater than 1.
The correct answers would be a and b
The numbers double: 3 x 2 = 6, 5 x 2 = 10, so the scale factor is 2.
Solve the system of equations.
y=x−5 y=x^2−2x−9
Enter your answers in the boxes.
( , ) or ( , )
sub the first equation into the next one
the expression -5t^2+ 40t predicts the height, in meters, of an object t seconds after a person launches it into the air. how many seconds will it take the object to hit the ground?
Answer:
8 seconds
Step-by-step explanation:
Set the equation equal to 0. You do this because h(0) means that there is no height of the object, and that then implies that the object is on the ground (where there is no height). Then factor using the quadratic formula:
[tex]0=-5t^2+40t[/tex]
Factor out the common constant and variable between the terms:
[tex]0=-5t(t-8)[/tex]
Solving for t, you get that t = 0 and t = 8. The t = 0 is indicative of the fact that at zero seconds (in other words before the object was launched) it was still on the ground. At t = 8, it had completed its parabolic travels and landed on the ground again.
The object launched in the air described by the quadratic equation [tex]-5t^2 + 40t[/tex] will hit the ground after 8 seconds. This is found by solving the quadratic equation [tex]0 = -5t^2 + 40t[/tex] for t.
The expression [tex]-5t^2 + 40t[/tex] predicts the height of an object at a given time after it is launched into the air. To find out when the object will hit the ground, we set the height to zero and solve for t. This results in a quadratic equation that we need to solve:
[tex]0 = -5t^2 + 40t[/tex]
Dividing each term by -5 to simplify, we get:
[tex]t^2 - 8t = 0[/tex]
We can factor this to:
t(t - 8) = 0
Setting each factor equal to zero gives us two solutions, t = 0 and t = 8. The solution t = 0 represents the object at launch. The solution t = 8 seconds represents the time when the object hits the ground.
Rewrite in a form that does not use exponents
Answer:
[tex]\boxed{18}[/tex]
Step-by-step explanation:
One of the rules of logarithms is ...
log(a^b) = b·log(a)
So ...
[tex]6\log_5{x^3}=6(3\log_5{x})=\bf{18}\log_5{x}[/tex]
Subject Algebra 1
If [tex]y=3x^{2} + x^{2} -5[/tex] and [tex]z=x^{2} -12[/tex] which polynomial is equivalent to [tex]2(y+z)[/tex]?
Answer:
[tex]10x^{2}-34[/tex]
Step-by-step explanation:
Because we are told equivalent expressions for y and z we can plug those in to 2(y+z).
[tex]2((3x^{2}+x^{2}-5)+(x^{2}-12))[/tex]
Then simplify by combining like terms of the expressions. Values ending in x^2 can be combined with each other.
[tex]2(5x^{2}-17)[/tex]
Now we can distribute the 2 by multiplying each value in the parentheses by 2.
[tex]10x^{2}-34[/tex]
A container contains 10 diesel engines. the company chooses 8 engines at random, and will not ship the container if any of the engines chosen are defective. find the probability that a container will be shipped even though it contains 2 defectives if the sample size is 8.
The probability that a container with 2 defective diesel engines will be shipped when 8 engines are chosen at random is 1/45, calculated using combinations in Mathematics.
Explanation:The subject of this question is Mathematics, specifically involving probability and combinatorics. A container has 10 diesel engines, with 2 being defective. The company will only ship the container if none of the randomly selected 8 engines are defective. The probability that the container is shipped can be found by considering the number of ways to choose 8 engines that are not defective out of the 10 engines, relative to the total number of ways to choose 8 engines out of 10 without any restrictions. Since there are 2 defective engines, there are 8 non-defective engines in total.
To calculate this, we use combinations: The number of ways to choose 8 non-defective engines from the 8 available is given by 8 choose 8 (which is 1 way), while the total number of ways to choose any 8 engines from the 10 is given by 10 choose 8. The probability is the ratio of these two numbers.
Using the combination formula which is n choose k = n! / (k!(n - k)!), we can calculate:
8 choose 8 = 1 (since 0! = 1 by definition)10 choose 8 = 10! / (8!(10 - 8)!) = 45The probability is therefore 1/45. Hence, the probability that a container will be shipped, even though it contains 2 defective engines, when 8 engines are chosen at random is 1/45.
What us the length of the altitude of the equilateral triangle below
Answer:
Height ( h) = 12 units.
Step-by-step explanation:
Given : An equilateral triangle with side = 8 √3.
To find : What us the length of the altitude of the equilateral triangle .
Solution : We have given equilateral triangle with side = 8 √3.
By taking the half triangle,
By the Pythagorean theorem :
(Hypotenuse)² = ( adjacent)² + (opposite)².
Plug the values Hypotenuse = 8√3 , adjacent = 4√3 , opposite = h.
(8√3)² = ( 4√3)² + (h)².
192 = 48 + (h)².
On subtracting by 48 both sides.
192 -48 = (h)².
144 = (h)².
On taking square root .
h = 12 .
Therefore, Height ( h) = 12 units.
The guy is right trust me mate
A recipe includes 4 cups of flour and 2/3 cup of brown sugar. Write the ratio of the amount of flour to the amount of brown sugar as a fraction in simplest form.
Answer:
6/1
Step-by-step explanation:
So we are looking for a ratio of flour which is 4 and brown sugar which is 2/3.
So we divide 4 by 2/3.
4÷2/3 since we are diving by a fraction we keep 4÷2/3 change 4*2/3 flip 4*3/2. So now we are multiplying 4 and 3/2.
So that is 4/1*3/2= 12/2 that simplies to 6/1
Answer:
⁶/₁
Step-by-step explanation:
4 cups of flour, ⅔ cup of brown sugar. The ratio of flour to brown sugar is:
4 / ⅔
We need to simplify this. First, write 4 as a fraction:
⁴/₁ / ⅔
To divide by a fraction, multiply by the reciprocal:
⁴/₁ × ³/₂
¹²/₂
⁶/₁
the volume of a sphere is 2,098 pi m^3 what is the surface area of the sphere to the nearest tenth?
1700
146.2
850
26,364
The surface area of the sphere to the nearest tenth is approximately 1700. Therefore the correct answer is option a.1700
To find the surface area of a sphere given its volume, we can use the formulas for the volume and surface area of a sphere:
1. Volume of a sphere: V = (4/3)πr³
2. Surface area of a sphere = A = 4πr²
Given:
V = 2098π
First, solve for the radius r using the volume formula
(4/3)πr³ = 2098π
Divide both sides by π:
(4/3)r³ = 2098
Multiply both sides by (3/4):
r³ = 2098 × (3/4)
r³ = 1573.5
Take the cube root of both sides to find r:
r ≈ 11.52
Now, use the radius to find the surface area A:
A = 4πr²
A = 4π(11.52)²
A = 4π × 132.7104
A = 530.8416π
Now, approximate the value:
A ≈ 530.8416 × 3.1416
A ≈ 1667.5
Rounding to the nearest tenth:
A ≈ 1667.5
Therefore, the surface area of the sphere to the nearest tenth is approximately 1700.
Complete Question:
The volume of a sphere is 2,098 pi m³ what is the surface area of the sphere to the nearest tenth?
a. 1700
b. 146.2
c. 850
d. 26,364
Answer:
1700 m^2
Step-by-step explanation:
The ratio of jakes money to joes money is 3:5. If joe gives jake $25, they will have an equal amount of money. How much money do they have altogether?
Answer: jake has 75 dollars and joe has 125 dollars
Step-by-step explanation:
3/5 = 75/125
joes gives 25 dollar
75 + 25 = 100
125 - 25 = 100
Ms.Holmes decided to water her plants.She had 2 1/2 gallons of water.She gave each plant 1/4 gallon of water.Which expression could be used to determine how many plants Ms.Holmes had?
To determine how many plants Ms. Holmes had, divide the total amount of water she had by the amount of water she gave each plant. The expression that can be used is (2 1/2 gallons) divided by (1/4 gallon per plant). Simplifying the expression gives the answer of 10 plants.
Explanation:To determine how many plants Ms. Holmes had, we can divide the total amount of water she had (2 1/2 gallons) by the amount of water she gave each plant (1/4 gallon per plant).
So, the expression that can be used is (2 ½ gallons) ÷ (1/4 gallon per plant). To divide a fraction by a fraction, we multiply the first fraction by the reciprocal of the second fraction. Therefore, the expression simplifies to (2 ½ gallons) × (4 gallons per plant).
Simplifying further, we get 10 gallons. This means that Ms. Holmes had 10 plants.
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Someone please help??
Answer:
x = 1
Step-by-step explanation:
The base is 3. The value of 3^x will be 3 only when x=1.
_____
The value of any exponential term will be equal to the base when the exponent is 1.
DO,h = (7,9) —> (14,18) the scale factor is ____.
Answer:
option B
2
Step-by-step explanation:
Given in the question
DO,h(7, 9) → (14, 18)
Formula to use
√(x1-x2)²+(y1-y2)²distance formula = √((7-0)²+(9-0)²) = √(7²+9²) = √130
distance formula = √((14-0)²+(18-0)²) = √(14²+18²) = 2√130
Scale factor
√130 = 2√130
d = 2d
Write the equation −2x−4y=−8 in slope-intercept form. Then graph the line described by the equation.
Answer:
y=-1/2x+2 start at y-intercept 2 and go down by 1 and right by 2
Step-by-step explanation:
The slope-intercept form of the equation −2x−4y=−8 is y = 0.5x + 2. Starting from the y-intercept (0,2), the graph of this line is drawn by moving a half unit up and one unit to the right from the intercept.
Explanation:The given mathematical equation is −2x−4y=−8. To convert this equation into slope-intercept form (y=mx+b), we want to isolate y. Start by dividing every term by -4, which simplifies the equation to 0.5x + y = 2. In this equation, the slope (m) is 0.5, and the y-intercept (b) is 2.
To graph this line, start by plotting the y-intercept (b) at the point (0, 2) on the y-axis. The slope (m) is 0.5, which we can interpret as a rise of 0.5 units for every run of 1 unit. So, from the y-intercept, move up half a unit and right one unit to plot your second point. Drawing a straight line through these points will give you the graph of your equation.
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Find x to the nearest hundredth.
7.73 cm
8.12 cm
20.46 cm
23.78 cm
Answer:
23.78
Step-by-step explanation:
CAH
x/25
cos18=x/25
25cos18=x
x=23.78
Answer:
23.78 cm
Step-by-step explanation:
Since, We know that,
In a right angle triangle,
[tex]cos \theta=\frac{B}{H}[/tex]
Where, B represents base adjacent to [tex]\theta[/tex]
H represents the hypotenuse adjacent to [tex]\theta[/tex]
Thus, by the given diagram,
[tex]cos 18^{\circ}=\frac{x}{25}[/tex]
[tex]\implies x = 25\times cos 18^{\circ}=23.7764129074\approx 23.78\text{ cm}[/tex]
Hence, Last option is correct.
Which of the following are possible rational roots of the polynomial function f(x)=5x^2-3x+3
You didn't post any option, but the ration roots theorem states that all possible rational roots of a polynomial come in the form
[tex]\pm\dfrac{p}{q}[/tex]
where p divides the constant term and q divides the leading term of the polynomial. So, in your case, p divides 3 (i.e. it is 1 or 3), and q divides 5 (i.e. it is 1 or 5).
So, the possible roots are
[tex]\pm 1,\quad \pm 3,\quad \pm\dfrac{1}{5},\quad \pm\dfrac{3}{5}[/tex]
For the record, this parabola has no real roots.
The possible rational roots of the polynomial function f(x) = 5x² - 3x + 3 are;
±1/5, ±3/5, ±1, ±3
We are given the polynomial function;
f(x) = 5x² - 3x + 3
The rational root theorem states that for a polynomial to have any rational roots, then the roots must be of the form;
±(Factors of coefficient of constant term/factors of the coefficient of the highest power)
Now, the coefficient of the highest power is 5 and the constant term is 3.
Factors of 5 = 1, 5
Factors of 3 = 1, 3
Thus, the possible rational roots are;
±1/5, ±3/5, ±1, ±3
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Find the cosine of angle p
Answer: last option
Step-by-step explanation:
You need to remember the identity:
[tex]cos\alpha=\frac{adjacent}{hypotenuse}[/tex]
You need to find the cosine of angle P, then, you can identify in the figure that:
[tex]adjacent=PR=21\\hypotenuse=PQ=29\\\alpha=P[/tex]
Therefore, the next step is substitute these values into [tex]cos\alpha=\frac{adjacent}{hypotenuse}[/tex], then you get:
[tex]cos(P)=\frac{21}{29}[/tex]
You can observe that this matches with the last option.
A survey showed that hamburgers are preferred to hot dogs 5 to 3. What percent of those surveyed prefer hamburgers
Final answer:
62.5% of those surveyed prefer hamburgers.
Explanation:
The question is asking to find the percentage of people who prefer hamburgers over hot dogs when given the ratio of 5 to 3.
To calculate this, you would add the two parts of the ratio together to get the total number of parts, which is 5 + 3 = 8.
The number of parts that represent hamburgers is 5.
To find the percentage, you would divide the hamburger part by the total number of parts and then multiply by 100.
This gives you (5/8) × 100, which equals 62.5%.
Therefore, 62.5% of those surveyed prefer hamburgers.
At a school supply store, binders cost $5 and pencil pouches cost $2. In one day, 42 of these items are sold for total sales of $144. Which system represents this (b represents binders and p represents pencil pouches), and how many binders were sold?
Answer:
Part 1) The system of equations that represent this situation is equal to
b+p=42 and 5b+2p=144
Part 2) 20 binders were sold
Step-by-step explanation:
Let
b -----> the number of binders sold
p -----> the number of pencil pouches sold
Part 1)
we know that
The system of equations that represent this situation is equal to
b+p=42
5b+2p=144
Part 2) How many binders were sold?
we have
b+p=42
p=42-b ------> equation A
5b+2p=144 -----> equation B
substitute equation A in equation B
5b+2(42-b)=144
5b+84-2b=144
3b=144-84
3b=60
b=20 binders
Please help me out with this
Answer:
[tex]\frac{9}{64}[/tex] π in²
Step-by-step explanation:
The area (A) of the circle is calculated using the formula
A = πr² ← r is the radius
here the diameter = [tex]\frac{3}{4}[/tex]
and radius is half the diameter, hence
r = [tex]\frac{3}{8}[/tex], so
A = π × ([tex]\frac{3}{8}[/tex] )² = [tex]\frac{9}{64}[/tex] π in²
The explicit rule for a sequence is an=5(−2)^n−1
What is the recursive rule for the sequence?
1) an=−2(an+1)
a1=5
2) an=−5(an+1)
a1=2
3) an=−2(an−1)
a1=5
4) an=−5(an−1)
a1=2
Answer:
3) [tex]a_n=-2a_{n-1}[/tex], [tex]a_1=5[/tex]
Step-by-step explanation:
Given that the explicit rule for a sequence is [tex]a_n=5(-2)^{n-1}[/tex].
Now we need to find about what is the recursive rule for the sequence and match with the given choices to find the correct choice.
1) [tex]a_n=-2a_{n+1}[/tex], [tex]a_1=5[/tex]
2) [tex]a_n=-5a_{n+1}[/tex], [tex]a_1=2[/tex]
3) [tex]a_n=-2a_{n-1}[/tex], [tex]a_1=5[/tex]
4) [tex]a_n=-5a_{n-1}[/tex], [tex]a_1=2[/tex]
Plug n=1 into given formula to get first term
[tex]a_n=5(-2)^{n-1}[/tex]
[tex]a_1=5(-2)^{1-1}=5(-2)^{0}=5(1)=5[/tex]
base of the exponent part is (-2) so that means we need to multiply -2 to the previous term to get nth term
Hence correct choice is: 3) [tex]a_n=-2a_{n-1}[/tex], [tex]a_1=5[/tex]
Answer:
3) an=−2(an−1)
a1=5
Step-by-step explanation:
NEED HELP ASAP 20 PTS PLEASE.
The graph of which is the following rational functions has a hole?
Answer:
see below
Step-by-step explanation:
The denominator quadratic of each of the rational functions has two real roots, so the rational function would ordinarily have two vertical asymptotes. If there is one vertical asymptote, it is because the other one has been canceled by a numerator factor, creating a "hole."
A graph of the first rational function shows it to have only one vertical asymptote, at x=3. The product of zeros of the denominator quadratic is the constant term, -12, so the other denominator zero must be at x=-4. That is where the hole is found. (See the graph in the second attachment.)
_____
Without a graphing calculator, you would determine the zeros of each quadratic, and identify the rational function that had numerator and denominator zeros that were the same.
[tex]f(x)=\dfrac{x^2+5x+4}{x^2+x-12}\\\\=\dfrac{(x+4)(x+1)}{(x+4)(x-3)} \qquad\text{has a common factor in numerator and denominator, a hole}[/tex]