Let the terms be a, ar, ar², ...
The sum to infinity being 27 means that:
[tex]\frac{a}{1-r}=27 \implies a=27(1-r)[/tex]
The sum of the first two terms being 15 means:
[tex]a+ar=15 \\ \\ (27-27r)+r(27-27r)=15 \\ \\ 27-27r+27r-27r^2 =15 \\ \\ 27r^2 -12=0 \\ \\ 27r^2 =12 \\ \\ r^2=\frac{12}{27}[/tex]
Since all the terms are positive, r>0.
[tex]\boxed{r=\frac{2}{3}} \\ \\ \implies a=27\left(1-\frac{2}{3} \right)=\boxed{9}[/tex]
find the area of 8in 5in 4in 4in 10in 6in
By decomposing the figure into simpler ones, we conclude that the area of the irregular figure is 108 square inches.
How to find the area of the irregular figure?
Here we have an irregular figure that can be decomposed into 3 simpler figures, these are:
A rectangle of 6 inches by 8 inches (the top one).A rectangle of 10 inches by 4 inches (the one in the middle).A rectangle of 5 inches by 4 inches (the one at the right).Remember that the area of a rectangle is the product between the two dimensions, so the areas of these 3 rectangles are:
a = 6in*8in = 48in^2
a' = 10in*4in = 40in^2
a'' = 5in*4in = 20in^2
Adding these areas we get:
48in^2 + 40in^2 + 20in^2 = 108 in^2
The area of the irregular figure is 108 square inches.
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explain how to write 50,000 using exponents
Explanation:
Place the decimal point between the first two digits to get 5.0
Then we must move the decimal point 4 spots to the right to get back to 50,000
So that's where the "4" comes in with the 10^4
This means 50,000 = 5 * 10^4 which is in scientific notation.
The answer in scientific notation is 5 times 10^4
The exponent of four tells you to multiply 10 by itself 4 times.
10 x 10 = 100 x 10 = 1000 x 10 = 10000
And 10000 times 5 is equal to 50,000.
Keith works at a garage. In the last week, the garage worked on 105 vehicles.
Out of these 35 were MOTS.
What percentage of the work was MOTS? Give your answer to 2 decimal places.
In the garage 33.33 percentage of work were MOTS.
Percentage is defined as per-cent, that is out of hundred. The percentage is calculated by dividing the acquired value by the total value and multiplying the result by 100.
Now in the given problem,
Total number of vehicles on which Keith worked is 105.
The number of MOTS among the 105 is 35.
Now we have to calculate the percentage of the MOTS out of a total 105 vehicles.
Let us use the formula=(number of MOTS ÷ total vehicles)×100
Therefore required percentage=[tex]\frac{35}{105}\times 100\%[/tex]
or, percentage=33.333...%
Therefore we can conclude that the total percentage of MOTS were approximately 33.33%(rounded to 2 decimal places).
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(100 points!) What is the slope-intercept form for the following equation?
x + 5y = 12
Questions:
x= 5y + 12
y= -1/5x + 12/5
x + 5y = 12x
5y= −x + 12
Answer:
the slope intercept form is 5y= -x+12
Suppose the scores on a test given to all juniors in a school district are normally distributed with a mean of 71 and a standard deviation of 5. Find the percent of juniors whose score is at least 71. % of juniors scored at least 71.
The percent of juniors whose score is at least 71 % is 69.15 %.
We are given that:
Test of the students are normally distributed.
Mean = μ = 71
Standard deviation = σ = 5
We need to find the percent of juniors whose score is at least 71 %.
Now, we know that:
z = ( x - μ) / σ
z = 100 - 71 / 5
z = 29 / 5
z = 5.8
So, percentage = 69.15 %
Therefore, we get that, the percent of juniors whose score is at least 71 % is 69.15 %.
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Find the least numbers which must be subtracted from the following number to make them perfect square
42029
Step-by-step explanation:
[tex] 3\sqrt{8} + \sqrt{114} [/tex]
A business has a budget of $720 to spend on their marketing campaign. Each month, x, the business spends $40 on different advertisements. The amount of money remaining in the budget
for marketing can be modeled by the function Mx)=-40x+720. Based on the graph of the linear function M(x) and the context of the problem, what is the domain?
O [10, 18]
O [10, 720]
0 [10, 8]
O [R]
A reasonable domain for this function is [0, 18)
Domain of a functionThe domain of a function is a value of the independent variable of the function for which it exists.
Given the equation below:
M(x)= -40x+720
Since the function represents the The amount of money remaining in the budget for marketing then the function f(x) must be greater than zero;
−40x + 720 > 0
-40x> -720
x < 720/40
x < 18
This shows that a reasonable domain for this function is [0, 18)
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Through (5,8); slope -2
X=-2×+18
How did we get 18
Answer:
y = -2x + 18
Step-by-step explanation:
Slope m = -2 with y = 8, x = 5
y = mx + b
8 = (-2)(5) + b
8 = -10 + b
b = 8 + 10 = 18
then y = -2x + 18
If the measure of A is 28 less than three times the measure of B. A and B are a linear pair. Find the measure of A and B.
Measure of linear pair A and B are 86 and 38 respectively.
Let the measure of B = x
And measure of A = 3x – 28
As A and B are linear pair. Therefore their sum will be equal to 180
That is numerically,
A + B = 180
Putting the required values of A and B we will get
3x – 28 + x = 180
4x = 180-28
4x = 152
x = 38
Hence the measure of A = 3*38 – 28 = 86
And the measure of B = x = 38
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2. Which statement correctly describes the
perpendicular bisector construction below?
The statement correctly describes the perpendicular bisector construction below The construction of JK to bisect GH: Option D is the correct answer
What is perpendicular bisector?The term perpendicular bisector refers to the line that divides another line into two equal parts, making an angle of 90 degrees. The term is made up of two words, perpendicular and bisector.
Perpendicular refers to angle 90 degrees, while the bisector refers to dividing into two equal parts.
In the picture, the main line is GH while JK is the line constructed to bisect the main line
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Whats (500 x 2) divided by 2?
Answer:
500
Step-by-step explanation:
500*2=1000
1000/2=500
SOLVE. y''+3y'+2y=4e^x cos3x
Solve the homogeneous equation
[tex]y'' + 3y' + 2y = 0[/tex]
Its characteristic equation is
[tex]r^2 + 3r + 2 = (r + 1) (r + 2) = 0[/tex]
with roots at [tex]r=-1[/tex] and [tex]r=-2[/tex], hence the characteristic solution is
[tex]y_c = C_1 e^{-x} + C_2 e^{-2x}[/tex]
For the nonhomogeneous equation, I'll use variation of parameters. We're looking for a solution of the form
[tex]y = u_1 y_1 + u_2 y_2[/tex]
to the equation
[tex]y'' + a(x) y'' + b(x) y = f(x)[/tex]
such that
[tex]\displaystyle u_1 = - \int \frac{y_2f(x)}{W(y_1,y_2)} \, dx[/tex]
[tex]\displaystyle u_2 = \int \frac{y_1 f(x)}{W(y_1,y_2)} \, dx[/tex]
The Wronskian [tex]W(y_1,y_2)[/tex] of the two fundamental solutions [tex]y_1=e^{-x}[/tex] and [tex]y_2=e^{-2x}[/tex] is
[tex]W(y_1,y_2) = \begin{vmatrix} y_1 & y_2 \\ {y_1}' & {y_2}' \end{vmatrix} = -e^{-3x}[/tex]
Then we have
[tex]\displaystyle u_1 = - \int \frac{e^{-2x} \cdot 4e^x \cos(3x)}{-e^{-3x}} \, dx = 4 \int e^{2x} \cos(3x) \, dx[/tex]
[tex]\displaystyle u_2 = \int \frac{e^{-x} \cdot 4e^x \cos(3x)}{-e^{-3x}} \, dx = -4 \int e^{3x} \cos(3x) \, dx[/tex]
Recall Euler's identity,
[tex]e^{(a+bi)t} = e^{at} (\cos(bt) + i \sin(bt))[/tex]
Then we have the general antiderivative
[tex]\displaystyle \int e^{(a+bi)t} \, dt = \frac1{a+bi} e^{(a+bi)t} + C = \frac{a-bi}{a^2+b^2} e^{(a+bi)t} + C[/tex]
Taking the real parts of both sides, we have
[tex]\displaystyle \mathrm{Re}\left\{\int e^{(a+bi)t} \, dt \right\} = \mathrm{Re}\left\{\frac{a-bi}{a^2+b^2} e^{(a+bi)t} + C\right\} \\\\ \int\,\mathrm{Re}\left\{e^{(a+bi)t}\right\} \, dt = \frac{e^{at}}{a^2+b^2} \mathrm{Re}\left\{(a-bi)(\cos(bt) + i \sin(bt))\right\} + C \\\\ \int e^{at} \cos(bt) \, dt = \frac{e^{at}}{a^2+b^2} (a\cos(bt)+b\sin(bt)) + C[/tex]
so that
[tex]\displaystyle u_1 = 4 \int e^{2x} \cos(3x) \, dx = \frac{4e^{2x}}{13} (2\cos(3x) + 3 \sin(3x))[/tex]
and
[tex]\displaystyle u_2 = -4 \int e^{3x} \cos(3x) \, dx = -\frac{2e^{3x}}3 (\cos(3x) + \sin(3x))[/tex]
We've found
[tex]y = u_1 y_1 + u_2 y_2[/tex]
[tex]\displaystyle y = \frac{4e^x}{13} (2\cos(3x) + 3 \sin(3x)) - \frac{2e^x}3 (\cos(3x) + \sin(3x))[/tex]
[tex]\displaystyle y = \frac2{39} e^x (5\sin(3x) - \cos(3x))[/tex]
Then the general solution to the differential equation is
[tex]\boxed{y(x) = C_1 e^{-x} + C_2 e^{-2x} + \frac2{39} e^x (5\sin(3x) - \cos(3x))}[/tex]
Answer 2 and 3. I’ll give Brainlyest and a lot of points.
Answer:
Please add the graph for these problems so I can solve your questions? Thank you!
Step-by-step explanation:
13. What value of b would make the function f (x)- Ibal stretch horizontally by a
factor of 4 and reflect across the y-axis?
Answer:
give me those points boy=o >:)
Step-by-step explanation:
While horizontal and vertical shifts involve adding constants to the input or to the function itself, a stretch or compression occurs when we multiply the parent function
f
(
x
)
=
b
x
by a constant
|
a
|
>
0
. For example, if we begin by graphing the parent function
f
(
x
)
=
2
x
, we can then graph the stretch, using
a
=
3
, to get
g
(
x
)
=
3
(
2
)
x
and the compression, using
a
=
1
3
, to get
h
(
x
)
=
1
3
(
2
)
x
.
Divide the following and then check by multiplying.
48,845
405
Answer:
The given Divisor = 405 and Dividend = 48849
The Quotient is 120 and the Remainder is 249
Hope it helps you
The variables y and x have a proportional relationship, and y = 12 when x = 5.
What is the value of y when x = 8?
Answer:
y = 19.2
Step-by-step explanation:
We know there is relationship between y and x, so according to the formula [tex]y = kx[/tex]
k = 12/5
we get one function y = 2.4x
when x = 8, input it into the function,
y = 2.4×8 = 19.2
Describe the end behavior of each polynomial.
(a) The leading coefficient is positive and the degree is odd, so:
As x approaches infinity, y approaches infinity.As x approaches negative infinity, y approaches negative infinity.(b) The leading coefficient is negative and the degree is even, so:
As x approaches infinity, y approaches negative infinity.As x approaches negative infinity, y approaches negative infinity.Juan and Mahdi are to run a race, but Juan can run 18 meters per second, whereas Mahdi can run 6 meters per second. If Mahdi has a 228 -meter head start, how long will it be before the faster runner catches the slower runner, if they begin at the same time? Round your answer to the nearest whole number.
Answer:
19 seconds
Step-by-step explanation:
You want to know the time it takes for Juan, running 18 m/s, to overtake Mahdi, running 6 m/s, if Mahdi has a 228 meter head start.
Closing timeThe time to close the gap between the runners will be ...
time = distance/speed
time = (228 m)/(difference in speeds) = (228 m)/(18 m/s -6 m/s)
time = (228 m)/(12 m/s) = (228/12) s = 19 s
It will take Juan 19 seconds to catch Mahdi.
__
Additional comment
The fastest human runner will not exceed 12.3 meters per second, especially over a 300+ meter distance.
On a website selling gifts, the total cost is given by the formula:
Total cost (£) = number of items × 12 + 5
Work out the total cost for 4 items.
Answer:
4x12 equals 48 therefore 48+5= 53......
Find the range please
Answer:
8
Step-by-step explanation:
The range of a data set in statistics is the difference between the largest and the smallest values. While range does have different meanings within different areas of statistics and mathematics, this is its most basic definition, and is what is used by the provided calculator. Using the same example:
2,10,21,23,23,38,38
38 - 2 = 36
The range in this example is 36. Similar to the mean, range can be significantly affected by extremely large or small values. Using the same example as previously:
2,10,21,23,23,38,38,1027892
The range, in this case, would be 1,027,890 compared to 36 in the previous case. As such, it is important to extensively analyze data sets to ensure that outliers are accounted for.
Answer: Range is all the values of the y-axis therefore, the range of the graph is {0,4,6,8}
Solve |5x+1|<-2 and write the solution in interval notation.
If you meant to say [tex]|5x+1| < -2[/tex], then there are no solutions.
This is because the |5x+1| is never negative. This applies to the result of any absolute value function. Absolute value represents distance on a number line. Negative distance is not possible.
---------------------------------------------------------------
If you meant to say [tex]|5x+1| \le 2[/tex], then the steps are shown below
[tex]|5x+1| \le 2\\\\-2 \le 5x+1 \le 2\\\\-2-1 \le 5x+1-1 \le 2-1\\\\-3 \le 5x \le 1\\\\-3/5 \le 5x/5 \le 1/5\\\\-3/5 \le x \le 1/5\\\\-0.6 \le x \le 0.2\\\\[/tex]
The interval notation would be [-3/5, 1/5] in fraction form
That is equivalent to [-0.6, 0.2] in decimal form.
Use square brackets to include each endpoint.
Find the solutions to x² = 28.
A. x=+7√2
OB. x=+2√/14
OC. x=+14√2
X
OD. x=+2√7
X
Answer:
x=+2√7
Step-by-step explanation:
Marty doesn't have any money saved and he does not want to wait to buy a car . He goes to the dealership and takes out a loan for the $15,000 car . His payments will be $ 500 per month for 3 years (36 months ) . How much will Marty pay for his $ 15,000 car ?
Marty will have to pay $18,000 for his $15,000 car.
While taking out a loan, the amount taken has to be repaid along with an interest. Thus, the amount paid back by the borrower is always greater than the initial amount that the borrower gets from the lender as a loan.
Here, we are given that Marty has to pay $500 per month for 3 years to repay the loan for his $15,000 car.
Thus, for one month he pays = $500
and also we know that 1 year = 12 months
⇒ 3 years = 12(3) = 36 months
Therefore, for 36 months Marty will have to pay,
500 × 36 = 18,000
Thus, Marty will have to pay $18,000 for his $15,000 car.
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Is (0, 3) a solution to the equation y=x+3? (1 point)
O yes
Ono
Answer: Yes, it is.
Step-by-step explanation:
given: x is 0 and y is 3. Plug in the equation and solve
y=x+3
3=0+3
3=3
Yes, it is a solution
Aviva's pay stub shows gross earnings of $662.30 for a week. Her regular rate of pay is $10.80 per hour for a 35 hour week and overtime is paid time and a half regular pay. How many Hours did she work
Number of hours Aviva's work for gross earnings of $662.30 for a week with regular rate of $10.80 per hour for 35 hour week and 1.5 of regular pay for overtime is 52.6hours.
Given,
Gross earning for week=$662.30
Regular rate per hour for 35hour week=$10.80
Overtime rate=1.5×(10.80)
For 'h' number of hours
(10.80×35)+(10.80×1.5) h=$662.30
⇒378 +16.20h=662.30
⇒h=284.30/16.20
⇒h≈17.6hours
Number of hours Aviva's work=35 +17.6
=52.6hours
Therefore, number of hours Aviva's work for gross earnings of $662.30 for a week with regular rate of pay $10.80 per hour for a 35 hour week and 1.5 of regular pay for overtime is 52.6hours.
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Erica is participating in a road race. The first part of the race is on a 5.2-mile-long straight road oriented at an angle of 25∘ north of east. The road then turns due north for another 6.0 mi to the finish line.
In miles, what is the straight-line distance from the starting point to the end of the race?
The distance between the starting point(initial) and the ending point(last) is known as displacement is 9.46miles.
What is displacement?
Displacement is defined as the change in position of an object. It's a vector volume and has a direction and magnitude. It's represented as an arrow that points from the starting position to the final position. For illustration- If an object moves from A position to B, also the object's position changes. Distance is a scalar volume that refers to" how important ground an object has covered" during its stir. Displacement is a vector volume that refers to" how far out of place an object is" it's the object's overall change in position. Displacement is a defense medium that involves an individual transferring negative heartstrings from one person or thing to another.In the first(1st) part of the race displacement of Erica is
d1 = 5.2 miles at 25(twenty-five) deg North of East
x(ax) and y-component of 1st(first) part of the race will be:
d1 = 5.2*cos 25(twenty five) deg i + 5.2*sin 25(twenty five) deg j
d1 = (4.71 i + 2.20 j) miles (this id d1)
In the second(2nd) part of the race
d2 = 6(six) miles towards north,
So
d2 = (6 j) miles (this is d2)
Now net(accurate) displacement of Erica will be, Using Vector addition(summation):
d = d1(4.71 i + 2.20 j) + d2(6j)
d = (4.71 i + 2.20 j) +(sum) (6 j)
d = 4.71 i +(sum) 8.20 j
So Magnitude of net(accurate) displacement will be:
|d| = [tex]\sqrt{4.71^2+8.20^2}[/tex]
|d| = 9.46 miles(nine.four-six) = distance between starting(initial) point and ending point
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1/2(x+4)=1/2x+1 is there a solution
Find the domain. Y= 1-7x / 8-|3x-15|
The domain is written in the form { x : x ≠ 7/3 }
Given data
Y= 1-7x / 8-|3x-15|
How to solve for the domainThe domain refers to the values of x, from the equation we solve for the possible values that will not make the equation undefined.
For the equation to be undefined the denominator should be zero hence we solve for the value of x that will make the equation, to have a denominator equal to zero and the domain will be any other values of x except the value that gives the denominator equal to zero,
The denominator is 8-|3x-15|
solving for the value of x that gives the equation that makes the equation equal to zero
8-|3x-15| = 0
3x - 15 = -8
3x = -8 + 15
3x = 7
x = 7/3
The domain is written in the form { x : x ≠ 7/3 }
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......................................
In general, a
person is 1% shorter in the evening than in the
morning. Use your height to write a conditional
that uses this fact.
The conditional statement is that; If generally a person is 1% shorter in the evening than in the morning, then my height is about 1.62326 m in the evening.
How to write conditional statements?
First, we will calculate how much is one percent of your height and then subtract that percentage from the total height to get your" height in the evening".
For example:
If my height is 1.64 m, then;
0.01 * 1.64 m = 0.0164 m
Step 2 is;
1.64m - 0.0164m = 1.62326 m
Now, using these height, let's write a conditional that uses the described fact:
Conditional:
If generally a person is 1% shorter in the evening than in the morning, then my height is about 1.62326 m in the evening.
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