Answer:
B. The set is linearly independent because at least one of the vectors is a multiple of another vector.
Step-by-step explanation:
A set of n vector of length n is linearly independent if the matrix with these vectors as column has none of zero determinant. The set of vectors is dependent if the determinant is zero. In the given question the vectors have no zero determinants therefore it is linearly independent.
in the diagram AB =AD and
Answer:
AC ≅ AE
Step-by-step explanation:
According to the SAS Congruence Theorem, for two triangles to be considered equal or congruent, they both must have 2 corresponding sides that are of equal length, and 1 included corresponding angle that is of the same measure in both triangles.
Given that in ∆ABC and ∆ADE, AB ≅ AD, and <BAC ≅ DAE, the additional information we need to prove that ∆ABC ≅ ADE is AC ≅ AE. This will satisfy the SAS Congruence Theorem. As there would be 2 corresponding sides that are congruent, and 1 corresponding angle in both triangles that are congruent to each other.
Answer:
A). AC ≅ AE
Step-by-step explanation: took test on edge
A deep-sea diver is in search of coral reefs.he finds a beautiful one at an elevation of -120 4/7feet. While taking pictures of the reef he catches sight of a manta ray. He swims up 25 3/7feet to check it out.what is the diver's new elevation?
Answer:-95 1/7 feet
Step-by-step explanation:
-120 4/7+25 3/7=-95 1/7 feet
please help Find: ∠x ∠a ∠b
9514 1404 393
Answer:
x = 22
a = 88°
b = 92°
Step-by-step explanation:
The angles marked with x-expressions are same-side interior angles, so are supplementary.
5x -18 +3x +22 = 180
8x +4 = 180
8x = 176
x = 22
Then the other marked angles are ...
a = 3x +22 = 3(22) +22 = 88 . . . degrees
b = 5x -18 = 5(22) -18 = 92 . . . degrees
Sven starts walking due south at 7 feet per second from a point 190 feet north of an intersection. At the same time Rudyard starts walking due east at 4 feet per second from a point 130 feet west of the intersection.
A. Write an expression for the distance d between Sven and Rudyard t seconds after they start walking.
B. What is the minimum distance between them?
C. When are Sven and Rudyard closest?
Answer: A. [tex]d=\sqrt{(190-7t)^2+(130-4t)^2}[/tex]
B. Minimum distance between them = 18.61 feet.
C. After 28.76 seconds Sven and Rudyard are closest.
Step-by-step explanation:
A) Let (0,0) be the intersection point.
Then, Initial Location of Sven (0,190).
Speed of Sven = 7 feet per second
Then, position of Sven after t seconds = (0,190-7t) [speed = distance x time]
Similarly, Initial position of Rudyard= (130,0)
Speed of Rudyard = 4 feet per second
Position after t seconds = (130-4t,0)
Distance d between Sven and Rudyard t seconds after they start walking:
[tex]d=\sqrt{(190-7t)^2+(130-4t)^2}[/tex]
B) Let [tex]d(t)=\sqrt{(190-7t)^2+(130-4t)^2}\\[/tex]
[tex]d'(t)=2(190-7t)(-7)+(2)(130-4t)(-4)\\\\=130t-3700[/tex]
Put d'(t)=0
[tex]130t-3700=0\\\\\Rightarrow\ t=\dfrac{3700}{130}\approx28.46\ sec[/tex]
Minimum distance :
[tex]d(28.46)=\sqrt{(190-7(28.46))^2+(130-4(28.46))^2}\\\\=\sqrt{346.154}\approx18.61\ feet[/tex]
Hence, the minimum distance between them = 18.61 feet.
c) After 28.76 seconds Sven and Rudyard are closest.
please answer asap. there are two pics :)
Answer:
[tex]\boxed{\sf A. \ 0.34}[/tex]
Step-by-step explanation:
The first triangle is a right triangle and it has one acute angle of 70 degrees.
We can approximate [tex]\sf \frac{WY}{WX}[/tex] from right triangle 1.
The side adjacent to 70 degrees is WY. The side or hypotenuse is WX.
The side adjacent to 70 degrees in right triangle 1 is 3.4. The side or hypotenuse is 10.
[tex]\sf \frac{3.4}{10} =0.34[/tex]
Click on the solution set below until the correct one is displayed.
Answer:
{ } or empty set.
Step-by-step explanation:
The solutions should be where the two lines intersect, but in this case, the parallel lines never intersect. That means that they have no solutions.
Hope this helps!
Answer:
{ } or empty set
Step-by-step explanation:
It's because these lines are parallel so they don't intersect to give you a coordinate.
A web page is accessed at an average of 20 times an hour. Assume that waiting time until the next hit has an exponential distribution. (a.) Determine the rate parameter λ of the distribution of the time until the first hit? (b.) What is the expected time between hits? (c.) What is the probability that t
Answer:
Step-by-step explanation:
Given that :
A web page is accessed at an average of 20 times an hour.
Therefore:
a. he rate parameter λ of the distribution of the time until the first hit = 20
b. What is the expected time between hits?
Let consider E(Y) to be the expected time between the hits; Then :
E(Y) = 1/λ
E(Y) = 1/20
E(Y) = 0.05 hours
E(Y) = 3 minutes
(c.) What is the probability that there will be less than 5 hits in the first hour?
Let consider X which follows Poisson Distribution; Then,
P(X<5) [tex]\sim[/tex] G(∝=5, λ = 20)
For 5 hits ; the expected time will be :
Let 5 hits be X
E(X) = ∝/λ
E(X) = 5/20
E(X) =1/4
E(X) = 0.25 hour
E(X) = 15 minutes
From above ; we will see that it took 15 minutes to get 5 hits; then
[tex]P(\tau \geq 0.25) = \int\limits^{\alpha}_{0.25} \dfrac{\lambda^{\alpha}}{\ulcorner^{\alpha}} t^{a\pha-1} \ e^{-\lambda t} \, dt[/tex]
[tex]P(\tau \geq 0.25) = \int\limits^{5}_{0.25} \dfrac{20^{5}}{\ulcorner^{5}} t^{5-1} \ e^{-20 t} \, dt[/tex]
[tex]\mathbf{P(\tau \geq 0.25) =0.4405}[/tex]
6th grade math help me, please. :)
Step-by-step explanation:
Hello there!!
no need to be panic we will help you, alright.
look solution in picture ok...
sorry for cutting in middle.
Hope it helps...
Which of the following is the standard form of y =3/7 x-1 a)3/7x-y=1 b) y-3/7x= - 1 c) 7y-3x= -7 d) 3x - 7y= 7
Answer:
d)
Step-by-step explanation:
the general form is ax + by = c
What is the base of the expression 11^12? A. 3 B. 11 C. 12 D. 21
An exponential has the base at the bottom, or the lower portion. Think of "basement" to help remember this. The exponent is the number up top, so 12 is the exponent.
Answer:
The person above me is correct ( :
Step-by-step explanation:
B
W varies inversely as the square root of x when x=4 w=4 find when x=25
Answer:
8/5
Step-by-step explanation:
w = k / √x
4 = k / √4
k = 8
w = 8 / √x
w = 8 / √25
w = 8/5
QUESTION 4 (10 MARKS)
A retired couple requires an annual return of $2,000 from investment of $20,000. There are 3
options available:
(A) Treasury Bills yielding 9%;
(B) Corporate bonds 11%;
Junk Bonds. 130%
How much should be invested in each to achieve their goal? Give 3 sets of options that can
achieve their goal.[10 Marks]
Answer:
(T, C, J) = (in dollars)
(10000, 10000, 0),
(15000, 4915.97, 84.03),
(18181.82, 1680.67, 137.51)
Step-by-step explanation:
There are a number of ways to approach this question. We have chosen an approach that determines the investments required to achieve interest rate targets.
__
For an overall interest rate of I, the proportion that must be invested at rate I1 < I < I2 is ...
proportion at I1 = (I2 -I)/(I2 -I1)
Similarly, the proportion that must be invested at I2 is what's left over. It can be computed similarly:
proportion at I2 = (I -I1)/(I2 -I1)
__
We want an overall interest rate of $2000/$20000 = 10%.
Given available interest rates of 9%, 11%, and 130%, we need to have investments at a rate lower than 10% and at a rate higher than 10%.
If we use only the options for 9% and 11% (no junk bonds), then we can compute ...
proportion at 9% = (11 -10)/(11 -9) = 1/2
proportion at 11% = (10 -9)/(11 -9) = 1/2
1st Option:
$10,000 in treasury bills; $10,000 in corporate bonds
__
Suppose we want to achieve a 13% return on our investments at 11% and 130%. Then the proportion invested at 9% will use this value for I2:
proportion at 9% = (13 -10)/(13 -9) = 3/4
Of the remaining 1/4 of the money, we can achieve a 13% return by mixing the investments like this:
proportion at 11% = (130 -13)/(130 -11) = 117/119
proportion at 130% = (13 -11)/(130 -11) = 2/119
2nd option:
$20,000 × 3/4 = $15,000 in treasury bills
$5000 × 117/119 = $4,915.97 in corporate bonds
The remaining amount, $84.03 in junk bonds
__
Let's suppose we want a 20% return on our investment in junk bonds and corporate bonds. Then the proportion of the money invested at 9% will be ...
proportion at 9% = (20 -10)/(20 -9) = 10/11
And the proportion at 11% will be ...
proportion at 11% = (130 -20)/(130 -11) = 110/119 . . . (of the remaining 1/11 of the funds)
3rd option:
$20,000 × 10/11 = $18,181.82 in treasury bills
$1,818.18 × 110/119 = $1,680.67 in corporate bonds
The remaining amount, $137.51 in junk bonds
_____
Additional comment
The most that could be invested in Junk Bonds is $165.29. If the remainder is invested in Treasury Bills, then the overall return will be $2000. (You could consider this to be a 4th option.)
the average temperature for one week in Alaska are as follows: 10, 6, 9, 2, 0,3. what is the mean of these tempartures ? show all work.
Answer:
5
Step-by-step explanation:
We know that we have to add all numbers then divide it by how many numbers there are. So, 10 + 6 + 9 + 2 + 0 + 3 = 30. 30/6 = 5.
Price of an item is reduced by 40% of its original price. A week later it’s reduced 20% of the reduced price. What’s the actual % of the reduction from the original price
Answer: 52%
Step-by-step explanation:
Let the original price be 100.
After 40% reduction, price will be 100 - 40% = 60
After further 20% reduction, price will be 60 - 20% = 48
%age = (cur val - orig. val ) / orig val x 100
= (48 - 100) / 100 x 100%
= -52
The actual percentage of reduction is 52%
The first reduction is given as:
[tex]r_1 = 40\%[/tex]
The second reduction is given as:
[tex]r_2 = 20\%[/tex]
Assume that the original price of the item is x.
After the first reduction of 40%, the new price would be:
[tex]New = x\times (1 -r_1)[/tex]
So, we have:
[tex]New = x\times (1 -40\%)[/tex]
[tex]New = x\times 0.6[/tex]
[tex]New = 0.6x[/tex]
After the second reduction of 20% on the reduced price, the new price would be:
[tex]New = 0.6x\times (1 -r_2)[/tex]
So, we have:
[tex]New = 0.6x\times (1 -20\%)[/tex]
[tex]New = 0.6x\times 0.8[/tex]
[tex]New = 0.48x[/tex]
Recall that the original price is x.
So, the actual reduction is:
[tex]Actual = \frac{x - 0.48x}{x}[/tex]
[tex]Actual = \frac{0.52x}{x}[/tex]
Divide
[tex]Actual = 0.52[/tex]
Express as percentage
[tex]Actual = 52\%[/tex]
Hence, the actual percentage of reduction is 52%
Read more about percentage change at:
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Use Newton's method with initial approximation x1 = −1 to find x2, the second approximation to the root of the equation x3 + x + 8 = 0. (Round your answer to four decimal places.) x2 =
Answer:
The second approximation to the root of the equation [tex]x^{3}+x+8 = 0[/tex] is -1.5000.
Step-by-step explanation:
The Newton's method is a numerical method by approximation that help find roots of a equation of the form [tex]f(x) = 0[/tex] with the help of the equation itself and its first derivative. The Newton's formula is:
[tex]x_{i+1} = x_{i} - \frac{f(x_{i})}{f'(x_{i})}[/tex]
Where:
[tex]x_{i}[/tex] - i-th approximation, dimensionless.
[tex]x_{i+1}[/tex] - (i+1)-th approximation, dimensionless.
[tex]f(x_{i})[/tex] - Function evaluated at the i-th approximation, dimensionless.
[tex]f'(x_{i})[/tex] - First derivative of the function evaluated at the i-th approximation, dimensionless.
The function and its first derivative are [tex]f(x) = x^{3}+x+8[/tex] and [tex]f'(x) = 3\cdot x^{2}+1[/tex], respectively. Now, the Newton's formula is expanded:
[tex]x_{i+1} = x_{i}-\frac{x_{i}^{3}+x_{i}+8}{3\cdot x_{i}^{2}+1}[/tex]
If [tex]x_{1} = -1[/tex], the value of [tex]x_{2}[/tex] is:
[tex]x_{2} = -1 - \frac{(-1)^{3}+(-1)+8}{3\cdot (-1)^{2}+1}[/tex]
[tex]x_{2} = -1.5000[/tex]
The second approximation to the root of the equation [tex]x^{3}+x+8 = 0[/tex] is -1.5000.
Answer:
-2.5000
Step-by-step explanation:
A rectangle has a width of 3/4 inches and a length of 9/10 inches. Another rectangle
is larger but still proportional to the first rectangle. It has a width of 30 inches and a length of 36 what proportion could model this situation
Answer:
Bigger size / smaller size = 40
Step-by-step explanation:
Notice that we
36 / (9/10) = 30 / (3/4) = 40
Therefore the proportion model would be
Bigger size / smaller size = 40
When the input is 4, the output of f(x) = x + 21 is
Answer:
25Step-by-step explanation:
When the input is 4, the output of f(x) = x + 21 is f(4).
Substitute x = 4 to f(x):
f(4) = 4 + 21 = 25
Answer:
25
Step-by-step explanation:
We can find the output by plugging in 4 as x into the function:
f(x) = x + 21
f(4) = 4 + 21
f(4) = 25
The prices for a loaf of bread and a gallon of milk for two supermarkets are shown below. Sue needs to buy bread and milk for her church picnic. At Supermarket A, she would pay $137.24. At Supermarket B, she would pay $140.04. Which of the following system of equations represents this situation?
Answer:
B. 3.19b + 4.59m = 137.24
3.49b + 4.39m = $140.04
Step-by-step explanation:
A B
Bread $3.19 $3.49
Milk $4.59 $4.39
Sue paid $137.24 in supermarket A
Sue paid $140.04 in supermarket B
Let
Price of bread A=$3.19
Price of bread B=$3.49
Price of milk A=$4.59
Price of milk B=$4.39
Quantity of Bread=b
Quantity of Milk=m
Pb=price of bread
Pm=price of milk
Qb=Quantity of bread
Qm=Quantity of milk
For each supermarket
Supermarket A Equation
PbQb + PmQm =$137.24
3.19b+ 4.59m = 137.24
Supermarket B Equation
PbQb + PmQm=$140.04
3.49b + 4.39m = $140.04
Combining both equations
3.19b + 4.59m = 137.24
3.49b + 4.39m = $140.04
Enter your answer in the box
____
Answer:
[tex]\boxed{2144}[/tex]
Step-by-step explanation:
The sum can be found by adding the parts:
[tex]\sum\limits_{n=1}^{32}{(4n+1)}=4\sum\limits_{n=1}^{32}{n}+\sum\limits_{n=1}^{32}{1}=4\cdot\dfrac{32\cdot 33}{2}+32\\\\= 2112+32=\boxed{2144}[/tex]
__
The sum of numbers 1 to n is n(n+1)/2.
Part F
I NEED HELP!
What is the geometric mean of the measures of the line segments A Dand DC? Show your work.
Answer:
AC2 = AB2 + BC2 ---> AC2 = 122 + 52 ---> AC = 13
AD / AB = AB / AC ---> AD / 12 = 12 / 13 ---> AD = 144/13
DC = AC - AD ---> DC = 13 - 144/13 ---> DC = 25/13
AD / DB = DB / DC ---> DB2 = AD · DC ---> DB2 = (144/13) · (25/13) ---> DB = 60/13
DB is the geometric mean of AD and DC.
Step-by-step explanation:
Whats the input value of f(x)=2x+5
Answer:
x
Step-by-step explanation:
f(x)=2x+5
Input: x
Output: f(x)
For i.e:
Input: 1
Output: f(1) = 2(1) + 5 = 2 + 5 = 7
The length l, width w, and height h of a box change with time. At a certain instant the dimensions are l = 3 m and w = h = 6 m, and l and w are increasing at a rate of 3 m/s while h is decreasing at a rate of 6 m/s. At that instant find the rates at which the following quantities are changing.
(a) The volume.
m3/s
(b) The surface area.
m2/s
(c) The length of a diagonal. (Round your answer to two decimal places.)
m/s
Answer:
a) The rate of change associated with the volume of the box is 54 cubic meters per second, b) The rate of change associated with the surface area of the box is 18 square meters per second, c) The rate of change of the length of the diagonal is -1 meters per second.
Step-by-step explanation:
a) Given that box is a parallelepiped, the volume of the parallelepiped, measured in cubic meters, is represented by this formula:
[tex]V = w \cdot h \cdot l[/tex]
Where:
[tex]w[/tex] - Width, measured in meters.
[tex]h[/tex] - Height, measured in meters.
[tex]l[/tex] - Length, measured in meters.
The rate of change in the volume of the box, measured in cubic meters per second, is deducted by deriving the volume function in terms of time:
[tex]\dot V = h\cdot l \cdot \dot w + w\cdot l \cdot \dot h + w\cdot h \cdot \dot l[/tex]
Where [tex]\dot w[/tex], [tex]\dot h[/tex] and [tex]\dot l[/tex] are the rates of change related to the width, height and length, measured in meters per second.
Given that [tex]w = 6\,m[/tex], [tex]h = 6\,m[/tex], [tex]l = 3\,m[/tex], [tex]\dot w =3\,\frac{m}{s}[/tex], [tex]\dot h = -6\,\frac{m}{s}[/tex] and [tex]\dot l = 3\,\frac{m}{s}[/tex], the rate of change in the volume of the box is:
[tex]\dot V = (6\,m)\cdot (3\,m)\cdot \left(3\,\frac{m}{s} \right)+(6\,m)\cdot (3\,m)\cdot \left(-6\,\frac{m}{s} \right)+(6\,m)\cdot (6\,m)\cdot \left(3\,\frac{m}{s}\right)[/tex]
[tex]\dot V = 54\,\frac{m^{3}}{s}[/tex]
The rate of change associated with the volume of the box is 54 cubic meters per second.
b) The surface area of the parallelepiped, measured in square meters, is represented by this model:
[tex]A_{s} = 2\cdot (w\cdot l + l\cdot h + w\cdot h)[/tex]
The rate of change in the surface area of the box, measured in square meters per second, is deducted by deriving the surface area function in terms of time:
[tex]\dot A_{s} = 2\cdot (l+h)\cdot \dot w + 2\cdot (w+h)\cdot \dot l + 2\cdot (w+l)\cdot \dot h[/tex]
Given that [tex]w = 6\,m[/tex], [tex]h = 6\,m[/tex], [tex]l = 3\,m[/tex], [tex]\dot w =3\,\frac{m}{s}[/tex], [tex]\dot h = -6\,\frac{m}{s}[/tex] and [tex]\dot l = 3\,\frac{m}{s}[/tex], the rate of change in the surface area of the box is:
[tex]\dot A_{s} = 2\cdot (6\,m + 3\,m)\cdot \left(3\,\frac{m}{s} \right) + 2\cdot (6\,m+6\,m)\cdot \left(3\,\frac{m}{s} \right) + 2\cdot (6\,m + 3\,m)\cdot \left(-6\,\frac{m}{s} \right)[/tex]
[tex]\dot A_{s} = 18\,\frac{m^{2}}{s}[/tex]
The rate of change associated with the surface area of the box is 18 square meters per second.
c) The length of the diagonal, measured in meters, is represented by the following Pythagorean identity:
[tex]r^{2} = w^{2}+h^{2}+l^{2}[/tex]
The rate of change in the surface area of the box, measured in square meters per second, is deducted by deriving the surface area function in terms of time before simplification:
[tex]2\cdot r \cdot \dot r = 2\cdot w \cdot \dot w + 2\cdot h \cdot \dot h + 2\cdot l \cdot \dot l[/tex]
[tex]r\cdot \dot r = w\cdot \dot w + h\cdot \dot h + l\cdot \dot l[/tex]
[tex]\dot r = \frac{w\cdot \dot w + h \cdot \dot h + l \cdot \dot l}{\sqrt{w^{2}+h^{2}+l^{2}}}[/tex]
Given that [tex]w = 6\,m[/tex], [tex]h = 6\,m[/tex], [tex]l = 3\,m[/tex], [tex]\dot w =3\,\frac{m}{s}[/tex], [tex]\dot h = -6\,\frac{m}{s}[/tex] and [tex]\dot l = 3\,\frac{m}{s}[/tex], the rate of change in the length of the diagonal of the box is:
[tex]\dot r = \frac{(6\,m)\cdot \left(3\,\frac{m}{s} \right)+(6\,m)\cdot \left(-6\,\frac{m}{s} \right)+(3\,m)\cdot \left(3\,\frac{m}{s} \right)}{\sqrt{(6\,m)^{2}+(6\,m)^{2}+(3\,m)^{2}}}[/tex]
[tex]\dot r = -1\,\frac{m}{s}[/tex]
The rate of change of the length of the diagonal is -1 meters per second.
15 POINTS+BRAINLIEST (Hurry now) A train goes past you in 10 seconds and goes past a 100 meter long bridge in 30 seconds. What is the length (in meters) and the speed (inm/s) of the train?
Answer:
speed=3.33m/s
Step-by-step explanation:
speed= distance÷time
3.33 m/s
length = ?
Answer:
Length : 50m
Speed : 5 m / s
im sorry if im too late :'(
your marksmanship score are 6 and 10 on two test . if you want average 9 on the tests , waht must your third score be?
Answer:
11
Step-by-step explanation:
To do this you would just multiply 9 by 3 so you get 27 and subtract 6+10 which is 16 from it and then you will get 11 and that is what you will need for your third score
The third score which must be added is 11.
What are average?The average can be calculated by dividing the sum of observations by the number of observations.
Average = Sum of observations/the number of observations
Given; count = 3 (there are three trials)
average = 9
9 = sum / 3
The sum = first score + second score + third score
The sum = 6 + 10 + third score
9 = (6+10+third score)/3
Then multiply both sides by 3 to remove the denominator
27 = 6 + 10 + third score
27 = 16 + third score
Now, subtract 16 from both sides to isolate the third score
11 = third score
Hence, the third score which must be added is 11.
Learn more about average here;
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Check whether these statements are wff or not:(a) (p˅q) ∧∼r
Answer:
It is a well formed formula
Step-by-step explanation:
1 - p,q,r are well formed formulas.
2 - [tex]p \ \lor \ q[/tex] is a well formed formula as well.
3 - [tex]\neg r[/tex] is a well formula as well
4 - [tex](\ p \ \lor \ q) \ \land \ \neg r[/tex] is a well formula as well.
Evaluate the integral. (Remember to use absolute values where appropriate. Use C for the constant of integration.) x4 18x2 4 x5 30x3 20x dx
Your integrand is missing some symbols. My best interpretation is the following integral:
[tex]I=\displaystyle\int\frac{x^4+18x^2+4}{x^5+30x^3+20x}\,\mathrm dx[/tex]
Decompose into partial fractions; we're looking for an expansion of the form
[tex]\dfrac{x^4+18x^2+4}{x^5+30x^3+20x}=\dfrac ax+\dfrac{bx^3+cx^2+dx+e}{x^4+30x^2+20}[/tex]
Now:
[tex]x^4+18x^2+4=a(x^4+30x^2+20)+(bx^3+cx^2+dx+e)x[/tex]
[tex]=(a+b)x^4+cx^3+(30a+d)x^2+ex+20a[/tex]
Matching up coefficients tells us that
[tex]\begin{cases}a+b=1\\c=0\\30a+d=18\\e=0\\20a=4\end{cases}\implies a=\dfrac15,b=\dfrac45,d=12[/tex]
so that
[tex]I=\displaystyle\frac15\int\frac{\mathrm dx}x+\frac45\int\frac{x^3+15x}{x^4+30x^2+20}\,\mathrm dx[/tex]
The integral is trivial:
[tex]\displaystyle\frac15\int\frac{\mathrm dx}x=\frac15\ln|x|+C[/tex]
For the second integral, notice that
[tex]\mathrm d(x^4+30x^2+20)=(4x^3+60x)\,\mathrm dx[/tex]
Distribute the 4 over the numerator, then substitute [tex]u=x^4+30x^2+20[/tex] and [tex]\mathrm du=(4x^3+60x)\,\mathrm dx[/tex]:
[tex]\displaystyle\frac15\int\frac{4x^3+60x}{x^4+30x^2+20}\,\mathrm dx=\frac15\int\frac{\mathrm du}u=\frac15\ln|u|+C=\frac15\ln(x^4+30x^2+20)+C[/tex]
So we have
[tex]I=\dfrac15\ln|x|+\dfrac15\ln(x^4+30x^2+20)+C[/tex]
and with some simplification,
[tex]I=\boxed{\ln\sqrt[5]{|x^5+30x^3+20x|}+C}[/tex]
A hotel rents 210 rooms at a rate of $ 60 per day. For each $ 2 increase in the rate, three fewer rooms are rented. Find the room rate that maximizes daily revenue.
Answer:
r=$14,400
The hotel should charge $120
Step-by-step explanation:
Revenue (r)= p * n
where,
p = price per item
n = number of items sold
A change in price leads to a change in number sold
A variable to measure the change in p and n needs to be introduced
Let the variable=x
Such that
p + x means a one dollar price increase
p - x means a one dollar price decrease
n + x means a one item number-sold increase
n - x means a one item number-sold decrease
for each $2 price increase (p + 2x) there are 3 fewer rooms are rented (n-3x)
know that at $60 per room, the hotel rents 210 rooms
r = (60 + 2x) * (210 - 3x)
=12,600-180x+420x-6x^2
=12,600+240x-6x^2
r=2100+40x-x^2
= -x^2 +40x+2100=0
Solve the quadratic equation
x= -b +or- √b^2-4ac / 2a
a= -1
b=40
c=2100
x= -b +or- √b^2-4ac / 2a
= -40 +or- √(40)^2 - (4)(-1)(2100) / (2)(-1)
= -40 +or- √1600-(-8400) / -2
= -40 +or- √ 1600+8400 / -2
= -40 +or- √10,000 / -2
= -40 +or- 100 / -2
x= -40+100/-2 OR -40-100/-2
=60/-2 OR -140/-2
= -30 OR 70
x=70
The quadratic equation has a maximum at x=70
p+2x
=60+2(30)
=60+60
=$120
r= (60 + 2x) * (210 - 3x)
={60+2(30)}*{(210-3(30)}
r=(60+60)*(210-90)
=120*120
=$14,400
Why should you find the least common denominator when adding or subtracting rational expressions?
Answer:
It is necessary to look for the least common denominator when one is trying to add or subtract rational expressions that do not have the same denominator.
Step-by-step explanation:
for example the denominator of the two addends are not the same. One has (x+2), the other (x-2).
You take one ball randomly from a bag with 10 yellow, 5 orange and 5 green balls. What is the probability that you take a yellow ball.
1
1/4
10/15
1/2
Answer:
1/2
Step-by-step explanation:
The probability of taking a yellow ball can be found by dividing the number of yellow balls over the total number of balls.
P(yellow ball)= yellow balls / total balls
There are 10 yellow balls. There are a total of 20 balls. There are 20 because there are 10 yellow, 5 orange, and 5 green. When 10, 5, and 5 are added, the result is 20.
yellow balls = 10
total balls= 20
P(yellow ball)= yellow balls / total balls
P(yellow ball)= 10/20
The fraction 10/20 can be simplified. Both the numerator( top number) and denominator (bottom number) can be evenly divided by 10.
P(yellow ball)= (10/10) / (20/10)
P(yellow ball)= 1/(20/10)
P(yellow ball)= 1/2
The probability of taking a yellow ball is 1/2.
find the area of the shaded region
Answer:
27 in²
Step-by-step explanation:
area of triangle (whole) = 1/2 x base x height
= 1/2 x 10 x 6
= 30 in²
area of small triangle = 1/2 x base x height
= 1/2 x 3 x 2
= 3 in²
area of shaded region = 30 in² - 3 in²
= 27 in²