Answer:
1.5
Step-by-step explanation:
Let the first term be a and the common ratio be r
ATQ, ar=50 and ar^3=112.5, divide these two. r^2=2.25, r=1.5
use undetermined coefficient to determine the solution of:y"-3y'+2y=2x+ex+2xex+4e3x
First check the characteristic solution: the characteristic equation for this DE is
r ² - 3r + 2 = (r - 2) (r - 1) = 0
with roots r = 2 and r = 1, so the characteristic solution is
y (char.) = C₁ exp(2x) + C₂ exp(x)
For the ansatz particular solution, we might first try
y (part.) = (ax + b) + (cx + d) exp(x) + e exp(3x)
where ax + b corresponds to the 2x term on the right side, (cx + d) exp(x) corresponds to (1 + 2x) exp(x), and e exp(3x) corresponds to 4 exp(3x).
However, exp(x) is already accounted for in the characteristic solution, we multiply the second group by x :
y (part.) = (ax + b) + (cx ² + dx) exp(x) + e exp(3x)
Now take the derivatives of y (part.), substitute them into the DE, and solve for the coefficients.
y' (part.) = a + (2cx + d) exp(x) + (cx ² + dx) exp(x) + 3e exp(3x)
… = a + (cx ² + (2c + d)x + d) exp(x) + 3e exp(3x)
y'' (part.) = (2cx + 2c + d) exp(x) + (cx ² + (2c + d)x + d) exp(x) + 9e exp(3x)
… = (cx ² + (4c + d)x + 2c + 2d) exp(x) + 9e exp(3x)
Substituting every relevant expression and simplifying reduces the equation to
(cx ² + (4c + d)x + 2c + 2d) exp(x) + 9e exp(3x)
… - 3 [a + (cx ² + (2c + d)x + d) exp(x) + 3e exp(3x)]
… +2 [(ax + b) + (cx ² + dx) exp(x) + e exp(3x)]
= 2x + (1 + 2x) exp(x) + 4 exp(3x)
… … …
2ax - 3a + 2b + (-2cx + 2c - d) exp(x) + 2e exp(3x)
= 2x + (1 + 2x) exp(x) + 4 exp(3x)
Then, equating coefficients of corresponding terms on both sides, we have the system of equations,
x : 2a = 2
1 : -3a + 2b = 0
exp(x) : 2c - d = 1
x exp(x) : -2c = 2
exp(3x) : 2e = 4
Solving the system gives
a = 1, b = 3/2, c = -1, d = -3, e = 2
Then the general solution to the DE is
y(x) = C₁ exp(2x) + C₂ exp(x) + x + 3/2 - (x ² + 3x) exp(x) + 2 exp(3x)
The average of two numbers is 5x. If one of the numbers is 2x + 3, find the other number.
Answer:
8x-3
Step-by-step explanation:
Average of 2 numbers means add the two numbers and divide by 2
(y+z)/2 = 5x
Let z = 2x+3
(y+2x+3)/2 = 5x
Multiply each side by 2
y+2x+3 = 10x
Subtract 2x from each side
y+3 = 10x-2x
y+3 = 8x
Subtract 3
y = 8x-3
The other number is 8x-3
WORTH 100 POINTS!
The function h(x) is quadratic and h(3) = h(-10) = 0. Which could represent h(x)?
1) h(x) = x2 - 13x - 30
2) h(x) = x2 - 7x - 30
3) h(x) = 2x2 + 26x - 60
4) h(x) = 2x2 + 14x - 60
Answer:
h(x) = 2x^2 +14x -60
Step-by-step explanation:
A quadratic is of the form
h(x) = ax^2 + bx +c
h(3) = h(-10) = 0
This tells us that the zeros are at x=3 and x = -10
We can write the equation in the form
h(x) = a( x-z1)(x-z2) where z1 and z2 are the zeros
h(x) = a(x-3) (x- -10)
h(x) = a(x-3) (x+10)
FOIL
h(x) = a( x^2 -3x+10x-30)
h(x) = a(x^2 +7x -30)
Let a = 2
h(x) = 2x^2 +14x -60
It means
zeros are 3 and -10
Form equation
y=x²-(3-10)x+(-10)(3)y=x²+7x-30Multi ply by 2
y=2x²+14x-60Option D
12) Find the angles between 0o and 360o where sec θ = −3.8637 . Round to the nearest 10th of a degree:
Please show all work
9514 1404 393
Answer:
105.0°, 255.0°
Step-by-step explanation:
Many calculators do not have a secant function, so the cosine relation must be used.
sec(θ) = -3.8637
1/cos(θ) = -3.8637
cos(θ) = -1/3.8637
θ = arccos(-1/3.8637) ≈ 105.000013°
The secant and cosine functions are symmetrical about the line θ = 180°, so the other solution in the desired range is ...
θ = 360° -105.0° = 255.0°
The angles of interest are θ = 105.0° and θ = 255.0°.
A.) V’ (-3,-5), K’ (-1,-2), B’ (3,-1), Z’(2,-5)
B.) V’(-4, 1), K’(-2, 4), B(2,5) Z’ (1, 1)
C.) V’ (-3,-4), K’(-1,-1) B’ (3,0), Z’(2,-4)
D.) V’ (-1,0), K’ (1, 3), B’(5,4), Z’(4,0)
Answer:
C
Step-by-step explanation:
this is a "translation" - a shift of the object without changing its shadow and size.
this shift is described by a "vector" - in 2D space by the x and y distances to move.
we have here (1, 0) - so, we move every point one unit to the right (positive x direction) and 0 units up/down.
therefore, C is the right answer (the x coordinates of the points are increased by 1, the y coordinate are unchanged).
If the terminal side of an angle (θ) goes through the point (4 , -3) what is (θ)?
Answer:
The family of directions of the given vector is represented by [tex]\theta = 323.130^{\circ} \pm 360\cdot i[/tex], [tex]\forall \,i\in \mathbb{N}_{O}[/tex].
Step-by-step explanation:
According to the given information, vector stands in the 4th Quadrant ([tex]x > 0[/tex], [tex]y < 0[/tex]) and direction of the vector ([tex]\theta[/tex]) in sexagesimal degrees, is determined by following definition:
[tex]\theta = 360^{\circ} - \tan^{-1} \left(\frac{|y|}{|x|} \right)\pm 360\cdot i[/tex], [tex]\forall \,i\in \mathbb{N}_{O}[/tex]
Please notice that angle represents a function with a periodicity of 360°.
If we know that [tex]x = 4[/tex] and [tex]y = -3[/tex], then the direction of the vector is:
[tex]\theta = 360^{\circ}-\tan^{-1}\left(\frac{|-3|}{|4|} \right)\pm 360\cdot i[/tex]
[tex]\theta = 323.130^{\circ} \pm 360\cdot i[/tex]
The family of directions of the given vector is represented by [tex]\theta = 323.130^{\circ} \pm 360\cdot i[/tex], [tex]\forall \,i\in \mathbb{N}_{O}[/tex].
Find the value of this expression
Answer:
[tex] \frac{(3) ^{2} + 3}{3 - 1} [/tex]
[tex] \frac{9 + 3}{3 - 1} [/tex]
[tex] \frac{12}{2} [/tex]
= 6
What is A11 for the geometric sequence 3,072, −1,536, -768, −384...?
Answer:
3
Step-by-step explanation:
The general formula of the series is 3072/(-2)^(n-1). A11=3072/(-2)^10=3
x = either 100 , 140 , or 120
help with 1 b please. using ln.
Answer:
[tex]\displaystyle \frac{dy}{dx} = \frac{1}{(x - 2)^2\sqrt{\frac{x}{2 - x}}}[/tex]
General Formulas and Concepts:
Pre-Algebra
Equality PropertiesAlgebra I
Terms/CoefficientsFactoringExponential Rule [Root Rewrite]: [tex]\displaystyle \sqrt[n]{x} = x^{\frac{1}{n}}[/tex]Algebra II
Natural logarithms ln and Euler's number eLogarithmic Property [Exponential]: [tex]\displaystyle log(a^b) = b \cdot log(a)[/tex]Calculus
Differentiation
DerivativesDerivative NotationImplicit DifferentiationDerivative Property [Multiplied Constant]: [tex]\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)[/tex]
Derivative Property [Addition/Subtraction]: [tex]\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)][/tex]
Basic Power Rule:
f(x) = cxⁿf’(x) = c·nxⁿ⁻¹Derivative Rule [Quotient Rule]: [tex]\displaystyle \frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}[/tex]
Derivative Rule [Chain Rule]: [tex]\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)[/tex]
Step-by-step explanation:
*Note:
You can simply just use the Quotient and Chain Rule to find the derivative instead of using ln.
Step 1: Define
Identify
[tex]\displaystyle y = \sqrt{\frac{x}{2 - x}}[/tex]
Step 2: Rewrite
[Function] Exponential Rule [Root Rewrite]: [tex]\displaystyle y = \bigg( \frac{x}{2 - x} \bigg)^\bigg{\frac{1}{2}}[/tex][Equality Property] ln both sides: [tex]\displaystyle lny = ln \bigg[ \bigg( \frac{x}{2 - x} \bigg)^\bigg{\frac{1}{2}} \bigg][/tex]Logarithmic Property [Exponential]: [tex]\displaystyle lny = \frac{1}{2}ln \bigg( \frac{x}{2 - x} \bigg)[/tex]Step 3: Differentiate
Implicit Differentiation: [tex]\displaystyle \frac{dy}{dx}[lny] = \frac{dy}{dx} \bigg[ \frac{1}{2}ln \bigg( \frac{x}{2 - x} \bigg) \bigg][/tex]Logarithmic Differentiation [Derivative Rule - Chain Rule]: [tex]\displaystyle \frac{1}{y} \ \frac{dy}{dx} = \frac{1}{2} \bigg( \frac{1}{\frac{x}{2 - x}} \bigg) \frac{dy}{dx} \bigg[ \frac{x}{2 - x} \bigg][/tex]Chain Rule [Basic Power Rule]: [tex]\displaystyle \frac{1}{y} \ \frac{dy}{dx} = \frac{1}{2} \bigg( \frac{1}{\frac{x}{2 - x}} \bigg) \bigg[ \frac{2}{(x - 2)^2} \bigg][/tex]Simplify: [tex]\displaystyle \frac{1}{y} \ \frac{dy}{dx} = \frac{-1}{x(x - 2)}[/tex]Isolate [tex]\displaystyle \frac{dy}{dx}[/tex]: [tex]\displaystyle \frac{dy}{dx} = \frac{-y}{x(x - 2)}[/tex]Substitute in y [Derivative]: [tex]\displaystyle \frac{dy}{dx} = \frac{-\sqrt{\frac{x}{2 - x}}}{x(x - 2)}[/tex]Rationalize: [tex]\displaystyle \frac{dy}{dx} = \frac{-\frac{x}{2 - x}}{x(x - 2)\sqrt{\frac{x}{2 - x}}}[/tex]Rewrite: [tex]\displaystyle \frac{dy}{dx} = \frac{-x}{x(x - 2)(2 - x)\sqrt{\frac{x}{2 - x}}}[/tex]Factor: [tex]\displaystyle \frac{dy}{dx} = \frac{-x}{-x(x - 2)^2\sqrt{\frac{x}{2 - x}}}[/tex]Simplify: [tex]\displaystyle \frac{dy}{dx} = \frac{1}{(x - 2)^2\sqrt{\frac{x}{2 - x}}}[/tex]Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentiation
Book: College Calculus 10e
Probability that a person is chosen at random
Answer:
152 / 370
Step-by-step explanation:
Total number of people
152+218 = 370
P( own a dog) = people said yes / total
= 152 / 370
what is the slope and point
Answer:
Step-by-step explanation:
These two cones are similar. What is the value of x?
Answer:
A
Step-by-step explanation:
Given that the cones are similar then corresponding dimensions are in proportion, that is
[tex]\frac{12}{2}[/tex] = [tex]\frac{3}{x}[/tex] ( cross- multiply )
12x = 6 ( divide both sides by 12 )
x = 0.5 → A
why infinity ( ) can’t be included in an inequality?
Answer:
Step-by-step explanation:
Because then the value on the other side will be unbounded by the infinity sign while expressing the answers on a number line.
please click thanks and mark brainliest if you like :)
We are given a weighted coin (with one side heads, one side tails), and we want to estimate the unknown probability pp that it will land heads. We flip the coin 1000 times and it happens to land heads 406 times. Give answers in decimal form, rounded to four decimal places (or more). We estimate the chance this coin will land on heads to
Answer:
0.4060
Step-by-step explanation:
To calculate the sample proportion, phat, we take the ratio of the number of preferred outcome to the total number of trials ;
Phat = number of times coin lands on head (preferred outcome), x / total number of trials (total coin flips), n
x = 406
n = 1000
Phat = x / n = 406/ 1000 = 0.4060
The estimate of the chance that this coin will land on heads is 0.406
Probability is the likelihood or chance that an event will occur.Probability = Expected outcome/Total outcomeIf a coin is flipped 1000 times, the total outcomes will 1000
If it landed on the head 406 times, the expected outcome will be 406.
Pr(the coin lands on the head) = 406/1000
Pr(the coin lands on the head) = 0.406
Hence the estimate of the chance that this coin will land on heads is 0.406
Learn more on probability here: https://brainly.com/question/14192140
Regina has 3 bags of marbles. There are 25 marbles in each bag. She wants to put an equal number of marbles into 5 bags. Which expression would show how many marbles can go in each bag?
Answer:
(3 × 25)/5 marbles can go in each bag
Explanation:
Number of bags Regina has = 3
Number of marbles in each bag = 25
So, total number of marbles = 3 × 25
Number of marbles in each bag, if divided equally into 5 bags = (3 × 25)/5
Further:
Solving the expression,
(3 × 25)/5
= 75/5
= 15
So, the each bag has 15 marbles if they are equally divided into 5 bags.
Answer:
(25 x 3) / 5
Step-by-step explanation:
you have to do 25 x 3 to get the total amount of marbles. Then you have to divide that by the amount of bags.
Addition prop of equality
subtraction prop of quality
multiplication prop of equality
Division prop of equality
simplifying
distrib prop
Illustrate the 7th pattern of the sequence of square numbers.
1,4,9,16,25,36,49,........
7th pattern =49.....
Answer:
1, 4, 9, 16, 25, 36, 49…................the 7 the pattern is 49
How to divided 245 by 70
Show your work
Answer:
Step-by-step explanation:
Hello!
2 4 5 ∟ 70
-2 1 3, 5
------------------------
3 5 0
3 5 0
- --------------------------------
0 0 0
A tour bus is traveling along a triangular path. The three straight lines form a right triangle. One leg of the triangle represents a distance of 8 miles. The other leg of the triangle is 4 miles shorter than the hypotenuse. What is the length of the hypotenuse of this triangle? Of the other leg?
Answer:
Hypotenuse=10 miles.
Short leg=6 miles.
Step-by-step explanation:
Set up triangle, leg 8 miles, hypotenuse x miles, short leg x-4 miles.Input into Pythagoras theorem.Simplify.The percent of data between z=0.23 and z = 1.27 is
(Round to two decimal places as needed.)
Answer:
0.40905 - 0.10204 = .30701 = 30.7 %
Step-by-step explanation:
0.23 0.40905
1.27 0.10204
Evaluate the expression: y – y ÷ 1 + x Use x = 7 and y = 3
Hi ;-)
[tex]x=7 \ and \ y=3\\\\y-y:1+x=3-3:1+7=3-3+7=0+7=\boxed7[/tex]
Again need help with these ones I don’t understand and they have to show work
Find the missing side length in the image below
Answer:
? = 5
Step-by-step explanation:
Recall: when 2 transversal lines cuts across 3 parallel lines, the parallel lines are divided proportionally by the transversals.
Therefore:
?/10 = 3/6
Cross multiply
?*6 = 3*10
?*6 = 30
Divide both sides by 6
? = 30/6
? = 5
Because of the relatively high interest rates, most consumers attempt to pay off their credit card bills promptly. However, this is not always possible. An analysis of the amount of interest paid monthly by a bank's Visa cardholders reveals that the amount is normally distributed with a mean of 27 dollars and a standard deviation of 9 dollars.
A. What proportion of the bank's Visa cardholders pay more than 29 dollars in interest?
B. What proportion of the bank's Visa cardholders pay more than 35 dollars in interest?
C. What proportion of the bank's Visa cardholders pay less than 14 dollars in interest?
D. What interest payment is exceeded by only 18% of the bank's Visa cardholders?
Answer:
a. 0.4129 = 41.29% of the bank's Visa cardholders pay more than 29 dollars in interest.
b. 0.1867 = 18.67% of the bank's Visa cardholders pay more than 35 dollars in interest.
c. 0.0742 = 7.42% of the bank's Visa cardholders pay less than 14 dollars in interest.
d. An interest payment of $35.2 is exceeded by only 18% of the bank's Visa cardholders.
Step-by-step explanation:
Normal Probability Distribution
Problems of normal distributions can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.
Mean of 27 dollars and a standard deviation of 9 dollars.
This means that [tex]\mu = 27, \sigma = 9[/tex]
A. What proportion of the bank's Visa cardholders pay more than 29 dollars in interest?
This is 1 subtracted by the p-value of Z when X = 29, so:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{29 - 27}{9}[/tex]
[tex]Z = 0.22[/tex]
[tex]Z = 0.22[/tex] has a p-value of 0.5871.
1 - 0.5871 = 0.4129
0.4129 = 41.29% of the bank's Visa cardholders pay more than 29 dollars in interest.
B. What proportion of the bank's Visa cardholders pay more than 35 dollars in interest?
This is 1 subtracted by the p-value of Z when X = 35, so:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{35 - 27}{9}[/tex]
[tex]Z = 0.89[/tex]
[tex]Z = 0.89[/tex] has a p-value of 0.8133.
1 - 0.8133 = 0.1867
0.1867 = 18.67% of the bank's Visa cardholders pay more than 35 dollars in interest.
C. What proportion of the bank's Visa cardholders pay less than 14 dollars in interest?
This is the p-value of Z when X = 14. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{14 - 27}{9}[/tex]
[tex]Z = -1.445[/tex]
[tex]Z = -1.445[/tex] has a p-value of 0.0742.
0.0742 = 7.42% of the bank's Visa cardholders pay less than 14 dollars in interest.
D. What interest payment is exceeded by only 18% of the bank's Visa cardholders?
This is the 100 - 18 = 82nd percentile, which is X when Z has a p-value of 0.82, so X when Z = 0.915.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]0.915 = \frac{X - 27}{9}[/tex]
[tex]X - 27 = 0.915*9[/tex]
[tex]X = 35.2[/tex]
An interest payment of $35.2 is exceeded by only 18% of the bank's Visa cardholders.
If per unit variable cost of a product is Rs.8 and fixed cost is Rs 5000 and it is sold for Rs 15 per unit, profit in 1000 units is.......
a.. rs 7000
b. rs 2000
c. rs 25000
d. rs 0
Answer:
a.. rs 7000
Because 15×1000=15000 it is SP when selling 1000units in the rate of Rs 15/unit& 8×1000=8000 this is cp when buying 1000 units in the rate of Rs 8/unit.
So,by formula of profit,
Rs (15000-8000)=Rs7000
Find the sum of ∑3/k=0 k^2
Answer:
[tex]14[/tex]
Step-by-step explanation:
Given
[tex]\displaystyle \sum_{k=0}^3k^2[/tex]
Let's break down each part. The input at the bottom, in this case [tex]k=0[/tex], is assigning an index [tex]k[/tex] at a value of [tex]0[/tex]. This is the value we should start with when substituting into our equation.
The number at the top, in this case 3, indicates the index we should stop at, inclusive (meaning we finish substituting that index and then stop). The equation on the right, in this case [tex]k^2[/tex], is the equation we will substitute each value in. After we substitute our starting index, we'll continue substituting indexes until we reach the last index, then add up each of the outputs produced.
Since [tex]k=0[/tex] is our starting index, start by substituting this into [tex]k^2[/tex]:
[tex]0^2=0[/tex]
Now continue with [tex]k=1[/tex]:
[tex]1^1=1[/tex]
Repeat until we get to the ending index, [tex]k=3[/tex]. Remember to still use [tex]k=3[/tex] before stopping!
Substituting [tex]k=2[/tex]:
[tex]2^2=4[/tex]
Substituting [tex]k=3[/tex]:
[tex]3^2=9[/tex]
Since 3 is the index we end at, we stop here. Now we will add up each of the outputs:
[tex]0+1+4+9=\boxed{14}[/tex]
Therefore, our answer is:
[tex]\displaystyle \sum_{k=0}^3k^2=0+1+4+9=\boxed{14}[/tex]
Answer:
14
Step-by-step explanation:
∑3/k=0 k^2
Let k=0
0^2 =0
Let k = 1
1^2 =1
Let k =2
2^2 = 4
Let k = 3
3^2 = 9
0+1+4+9 = 14
if point B is the midpoint of points A and C, find the value of x and AC. AB= 5x - 2, BC= 9x -10
9514 1404 393
Answer:
x = 2AC = 16Step-by-step explanation:
The midpoint divides the segment into two equal lengths:
AB = BC
5x -2 = 9x -10
8 = 4x
2 = x
AB = 5(2) -2 = 8
AC = 2AB = 2(8) = 16
8x=3x²-1 plz help me show your work
Answer:
Step-by-step explanation:
3 times 8= 24 • 24 = 576 - 1 =575
or
3•8=24•2=48-1=47
not sure
Answer:
The answer is [tex]x=\frac{4(+-)\sqrt{19} }{3}[/tex] in exact form or [tex]x=2.7863[/tex], [tex]x=-0.1196[/tex] in decimal form.
Step-by-step explanation:
To solve this equation, start by moving all expression to the left side of the equation, which will include subtracting [tex]3x^2[/tex] and adding 1 to both sides of the equation. The equation will look like [tex]8x-3x^2+1=0[/tex].
Then, use the quadratic formula to find the solutions to the equation. The quadratic formula looks like [tex]\frac{-b(+-)\sqrt{b^2-4ac} }{2a}[/tex].
For this problem, the quadratic variables are as follows:
[tex]a=-3\\b=8\\c=1[/tex]
The next step is to substitute the values [tex]a=-3[/tex], [tex]b=8[/tex], and [tex]c=1[/tex] into the quadratic formula and solve for x. The quadratic formula will look like [tex]\frac{-8(+-)\sqrt{8^2-4(-3)(1)} }{2*-3}[/tex]. To simplify the equation, start by simplifying the numerator, which will look like [tex]x=\frac{-8(+-)2\sqrt{19} }{2*-3}[/tex]. Then, multiply 2 by -3 and simplify the equation, which will look like [tex]x=\frac{4(+-)\sqrt{19} }{3}[/tex]. The final answer is [tex]x=\frac{4(+-)\sqrt{19} }{3}[/tex] in exact form. In decimal form, the final answer is [tex]x=2.7863[/tex], [tex]x=-0.1196[/tex].
Which equation is represented by the graph below?
Answer:
Hello,
Answer C
Step-by-step explanation:
Since ln(1)=0
if x=1 then y=4 ==> y=ln(x)+4
y=ln(x) is translated up for 4 units.
