Answer:
[tex]\large\boxed{x=\dfrac{-3+\sqrt{40+10\sqrt{10}}}{2}}[/tex]
Step-by-step explanation:
[tex]2\log x=3-2\log(x+3)\\\\Domain:\ x>0\ \wedge\ x+3>0\to x>-3\\\\D:x>0\\============================\\2\log x=3-2\log(x+3)\qquad\text{add}\ 2\log(x+3)\ \text{to both sides}\\\\2\log x+2\log(x+3)=3\qquad\text{divide both sides by 2}\\\\\log x+\log(x+3)=\dfrac{3}{2}\qquad\text{use}\ \log_ab+\log_ac=\log_a(bc)\\\\\log\bigg(x(x+3)\bigg)=\dfrac{3}{2}\qquad\text{use the de}\text{finition of a logarithm}\\\\x(x+3)=10^\frac{3}{2}\qquad\text{use the distributive property}[/tex]
[tex]x^2+3x=10^{1\frac{1}{2}}\\\\x^2+3x=10^{1+\frac{1}{2}}\qquad\text{use}\ a^n\cdot a^m=a^{n+m}\\\\x^2+3x=10\cdot10^\frac{1}{2}\qquad\text{use}\ \sqrt[n]{a}=a^\frac{1}{n}\\\\x^2+3x=10\sqrt{10}\qquad\text{subtract}\ 10\sqrt{10}\ \text{from both sides}\\\\x^2+3x-10\sqrt{10}=0\\\\\text{Use the quadratic formula}\\\\ax^2+bx+c=0\\\\x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\\\\a=1,\ b=3,\ c=-10\sqrt{10}\\\\b^2-4ac=3^2-4(1)(-10\sqrt{10})=9+40\sqrt{10}\\\\x=\dfrac{-3\pm\sqrt{40+10\sqrt{10}}}{2(1)}=\dfrac{-3\pm\sqrt{40+10\sqrt{10}}}{2}\\\\x=\dfrac{-3-\sqrt{10+10\sqrt{10}}}{2}\notin D[/tex]
Joe is riding his bike home. He begins 10 miles away from his house and is riding home at a speed of 0.25 miles per minute. Which function f(m) represents the distance he is away from his house given how many minutes m he has been riding?
Select one:
a. f(m)=10m−0.25m
b. f(m)=10−0.25m
c. f(m)=0.25+10m
d. f(m)=10+0.25m
The answer is:
The correct answer is:
d. [tex]f(m)=10+0.25*m[/tex]
Why?From the statement we know that he begins at 10 miles away from his house riding at a speed of 0.25 miles per hour (h)
We have that:
[tex]x_{f}=x_{o}+v*t[/tex]
Where,
[tex]x=FinalPosition\\x_{o}=InitialPosition\\v=speed\\t=time[/tex]
Now, using the given information we have that=
[tex]x_{f}=x_{o}+v*t[/tex]
[tex]x_{f}=10miles+0.25mph*m[/tex]
We have that:
[tex]x_{f}=f(m)[/tex]
So,
[tex]f(m)=10miles+0.25mph*m[/tex]
Hence, we have that the correct answer is:
d. [tex]f(m)=10+0.25*m[/tex]
Have a nice day!
Final answer:
The function that represents Joe's distance from his house after riding for m minutes is f(m) = 10 - 0.25m. Hence, the answer is B.
Explanation:
The correct function to represent the distance Joe is from his house after riding for m minutes is f(m) = 10 - 0.25m. This is because Joe starts 10 miles away from his house and rides towards it, reducing the distance by 0.25 miles every minute.
To form the function, you need to start with the initial distance, which is 10 miles. Since Joe is moving closer to his house, the distance decreases over time. Therefore, for every minute m that Joe rides, he covers 0.25 miles. Thus, the distance from home after m minutes is the initial distance minus the product of the speed and time, which gives us f(m) = 10 - 0.25m.
Which system of inequalities is shown in the graph
Answer:
Option D
Step-by-step explanation:
step 1
Find the equation of the inequality A (quadratic equation)
The quadratic equation is
[tex]y=x^{2}-3x[/tex]
we know that
The solution of the inequality A is the shaded area above the solid line of the quadratic equation
so
The inequality is equal to
[tex]y\geq x^{2}-3x[/tex]
step 2
Find the equation of the inequality B (linear equation)
The linear equation is
[tex]y=-x+3[/tex]
we know that
The solution of the inequality B is the shaded area below the solid line of the linear equation
so
The inequality is equal to
[tex]y\leq -x+3[/tex]
Answer: it’s c
Step-by-step explanation:
given the functiom y=x^4 -8x^2+16. on which intervals is the function increasing
Answer:
[tex]x\in (-2,0)\cup (2,\infty)[/tex]
Step-by-step explanation:
Find the derivative of the function [tex]y=x^4 -8x^2 +16:[/tex]
[tex]y'=4x^3 -8\cdot 2x\\ \\y'=4x^3 -16x[/tex]
The function is increasing when [tex]y'>0,[/tex] so solve the inequality
[tex]4x^3-16x>0\\ \\4x(x^2-4)>0\\ \\4x(x-2)(x+2)>0\\ \\x\in (-2,0)\cup (2,\infty)[/tex]
You can see from the graph that the function increases for [tex]x\in (-2,0)\cup (2,\infty)[/tex]
Final answer:
To determine the intervals where the function y = x⁴ - 8x² + 16 is increasing, calculate the first derivative, find its critical points, and analyze the sign changes. The function is increasing on the intervals (-∞, -2), (0, 2), and (2, ∞).
Explanation:
Intervals Where the Function is Increasing
To determine on which intervals the function y = x⁴ - 8x² + 16 is increasing, we need to analyze the function's first derivative. The first derivative, y', indicates the slope of the tangent to the function at any point. When y' > 0, the function is increasing. Let's find the first derivative of the given function:
Take the derivative of the function y with respect to x: y' = 4x³ - 16x.
Set the derivative equal to zero to find critical points: 4x³ - 16x = 0.
Factor out the common term (4x): 4x(x² - 4) = 4x(x + 2)(x - 2) = 0.
Solve for x to find critical points: x = 0, x = -2, and x = 2.
Analyze the sign of y' around these points to determine where the function is increasing and decreasing.
Through analyzing the signs, you will find that the function is increasing on the intervals (-∞, -2), (0, 2), and (2, ∞).
Note that these intervals are determined by looking at the sign changes of the first derivative around the critical points and selecting the intervals where the derivative is positive.
What is the quotient of 78,600 + 12?
6,300
6.460
6,550
6.630
bious Question
Question 15 (Not Answer
The quotient of 78,600 and 12 is 6,550
Answer:
6,550
Step-by-step explanation:
What is the value of x in this figure? 282√ 28 143√ 283√ A right triangle with one of the acute angles labeled 30 degrees. The hypotenuse is labeled x. The leg across from 30 degree angle is labeled 14.
Using the law of sines:
Sin (angle) = opposite leg / hypotenuse
Sin(30) = 14 /x
x = 14 * sin(30)
x = 28
Answer:
the answer is 28
Step-by-step explanation:
Amy has a box containing 6 white, 4 red, and 8 black marbles. She picks a marble randomly. It is red. The second time, she picks a white marble. In the third attempt, Amy picks a white marble again. Amy has not replaced the marbles she picked.
not really sure what the question is but if your question is but if ur question is asking what the possiblity is, the answer should be a possiblity of 8/15 for picking a black marble. If Amy picks a black marble in the fourth attempt, the probability of the fifth attempt she will pick a red or a black marble is 5/7
Write fifty-three
hundredths in
standard form.
Answer:
O.53 is the answer.
Step-by-step explanation:
Since standard form is more like number form, fifty three hundredths would be 0.53 in standard form.
I need help. ASAP!
Divide.
(3b^3 – 10b^2 + 4) ÷ (3b – 1)
Step-by-step explanation:
try surfing how to divide a function by the long division method
The balance sheet contains these three elements of a business
Answer:
The balance sheet consists of three major elements: assets, liabilities and owners' equity. The object of the statement is to prove true the accounting equation, "Asset = Liabilities + Owner's Equity."
Hope this helps!! :)
What is 3 log y -2 log x -4 log z written as a single logarithm
Use rules of logarithms to condense. log ( y 3 x 2 z 4 )
The given expression 3 log y - 2 log x - 4 log z can be combined into one logarithm using rules of logarithms. We obtain the single logarithm: [tex]log [(y^3)/(x^2*z^4)][/tex]. Feel free to apply these rules to other similar problems.
Explanation:The problem asks you to write the given expressions 3 log y - 2 log x - 4 log z as a single logarithm. From the rules of logarithms, we know that:
log [tex]a^n[/tex] = n log a (i.e., the power rule of logarithms).log (a/b) = log a - log bUsing these rules, our expression simplifies as such:
3 log y - 2 log x - 4 log z
is equivalent to:
[tex]log(y^3) - log(x^2) - log(z^4)[/tex]
This can further be combined, using the second rule into:
[tex]log [(y^3)/(x^2*z^4)][/tex]
This method can be used to simplify any given logarithmic expression.
Learn more about Logarithm here:https://brainly.com/question/33159023
#SPJ2
A teacher is randomly calling on students in a class. If there are 6 girls and 6 boys in the class, what is the theoretical probability that the first 3 people called on are all girls?
The probability of calling three girls consecutively in a mixed class will be 1/22.
There are 12 students in total.
The probability of the first student being a girl is 6 / 12 = 1/2.
After one girl has been chosen, there are now 5 girls left and a total of 11 students.
So, the probability of the second student being a girl is 5 / 11.
With one more girl removed, there would be 4 girls left and 10 students in total.
The probability of the third student being a girl then is 4 / 10 = 2/5.
To find the overall probability of all three events happening in sequence, we multiply the probabilities of each event:
Probability = (1/2) x (5/11) x (2/5)
= 1/22
The theoretical probability that the first 3 students called are all girls is: [tex]\[\boxed{\frac{1}{11}}\][/tex].
Given a class of 12 students, consisting of 6 girls and 6 boys, we are to determine the probability that the first 3 students called are all girls.
First, we calculate the total number of ways to select any 3 students out of 12. This is given by the combination formula:
[tex]\[\binom{12}{3} = \frac{12!}{3!(12-3)!} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220\][/tex]
Next, we calculate the number of ways to select 3 girls out of the 6 available girls:
[tex]\[\binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20\][/tex]
The theoretical probability that the first 3 students called are all girls is the ratio of the number of ways to choose 3 girls to the total number of ways to choose any 3 students. Thus, the probability \(P\) is:
[tex]\[P = \frac{\binom{6}{3}}{\binom{12}{3}} = \frac{20}{220} = \frac{1}{11}\][/tex]
Please answer right away
Answer:
20 miles with an error margin of ± 8 miles
Step-by-step explanation:
The margin of error of a result is the range in which an error can vary. To find the margin of error between both distances we have to
28-12 = 16, that is, the variation of the result has a range of 16 miles. So we will look for the midpoint of both distances
(X2-X1)/2+X1=(28-12)/2+12=16/2+12=8+12=20
So from this midpoint the value can vary between 8 points below and 8 points above that would cover the difference of 16 miles that we observed at the beginning
In this way, the correct answer is 20 miles with an error margin of ± 8 miles
Done
To make a profit, Ethan knows that his prime cost of producing a part must be no more than $189.27. It takes 2 hours for a machinist to make the part. The direct cost of materials is $98.62. To the nearest dollar, what can he afford to spend on direct labor costs per hour?
Answer:
$45/h
Step-by-step explanation:
So, we have a total maximum cost for the finished product and we have the raw material costs... but we need to figure out the maximum labor cost (for 2 hours).
We can express this as follows (remember it takes 2 hours of labor):
98.62 + 2x <= 189.27
2x <= 90.65
x <= 45.325
Rounded to the nearest dollar: $45
The maximum cost Ethan can afford in labor is $45/h for the 2 hours machining of the parts.
Please really need help on this
Answer:
x = 8
3rd choice
Step-by-step explanation:
https://mathbitsnotebook.com/Geometry/Trigonometry/TGTrigSineCosine.html
an equilateral triangle has a side length of 10 units. what is its area?
Answer:
Area of equilateral triangle = 43.3 units^2
Step-by-step explanation:
An equilateral triangle is a triangle in which all three sides of triangle are equal.
So,Length = 10 units
Area of triangle = [tex]\frac{\sqrt{3} }{4}*(side)^2[/tex]
= [tex]\frac{\sqrt{3} }{4}*(10)^2[/tex]
= [tex]\frac{\sqrt{3} }{4}*(100)[/tex]
= [tex]43.3units^2[/tex]
So, Area of equilateral triangle = 43.3 units^2.
The answer is:
The area of the triangle is equal to [tex]A=43.30units^{2}[/tex]
Why?To calculate the area of an equilateral triangle, we need to use the following formula:
[tex]A=\frac{s^{2} *\sqrt{3}}{4}[/tex]
Now, we are given an equilateral triangle, so we know that all of its sides are equal to 10 units.
So, calculating the area we have:
[tex]A=\frac{s^{2} *\sqrt{3} }{4}\\\\A=\frac{(10units)^{2} *\sqrt{3} }{4}\\\\A=\frac{100units^{2} *\sqrt{3} }{4}=43.30units^{2}[/tex]
Have a nice day!
What is the slope of the line 2x – 5y = 12 ?
Answer:
2x−5y=12
y = 2/5x + −12/5
Answer is x = 5/2y + 6 because....
First, Add 5y to both sides.
2x − 5y + 5y = 12 + 5y
Then, Divide both sides by 2.
[tex]\frac{2x}{2} = \frac{5y+12}{2}[/tex]
Therefor, your answer is going to be x = 5/2y + 6
* Hopefully this helps:) Mark me the brainliest:)!!!
Answer:
2/5
Step-by-step explanation:
We are given the following equation of a line and we are to tell the slope of the line:
[tex]2x-5y=12[/tex]
We know that the standard equation of a line is given by:
[tex] y = m x + c [/tex] where m is the slope of the line.
So re-writing the given equation in the standard form:
[tex]y=\frac{2}{5} x-\frac{12}{5}[/tex]
Therefore, 2/5 is the slope of the line.
7. What should I expect if you play 36 times
8. How many preferred 2 door cars
DO NOT NEED STEPS
Answer:
7: 9 times 8: 108
Answer:
7. 9 times
8. 108 people
Step-by-step explanation:
7. The probability of winning is 25%
So, if a person plays 36 times, the probability will be:
25% of 36
=> 36 * 25/100
=> 36 * 0.25
=> 9 times
So, option 3 is correct .
8. Total number of people asked are 300
and
The percentage of people who prefer two-door cars are 36%
So, the number of people who prefer two-door car are:
300 * 36/100
=> 300 * 0.36
=> 108 people
So, option 1 is correct ..
3/8 *5/6 pls help asap
Answer:
5/16
Step-by-step explanation:
It's a little easier when you write the problem like this:
3 5
----- · -----
8 6
Now reduce that 3/6 to 1/2:
1 5
----- · ----- = 5/16
8 2
Hello There!
We are given the fraction 3/8 multiplied by 5/6.
Step 1. First, we multiply across so we multiply 3*5 which is 15 and then we multiply 8*6 which equals 48.
Step 2. Both of these can be divided by 3 so we can put the fraction that we already have in simplest form. Dividing 15 by 3 equals 5 and dividing 48 by 3 equals 16.
Final step. We now have our simplified fraction which is 5/16.
Have a great day!
Be safe!
TheBlueFox
The circumference of a pizza is 81 in. What is the radius?
If the circumference of a pizza is 81 inches, the radius would be 12.89155 inches.
Please help me idk this
Answer:
[tex]\large\boxed{Surface\ Area=63\ cm^2}[/tex]
Step-by-step explanation:
We have
square with sides s = 3cm
four triangles with base b = 3cm and height h = 9cm.
The formula of an area of a square:
[tex]A_\square=s^2[/tex]
The formula of an area of a triangle:
[tex]A_\triangle=\dfrac{bh}{2}[/tex]
Substitute:
[tex]A_\square=3^2=9\ cm^2\\\\A_\triangle=\dfrac{(3)(9)}{2}=\dfrac{27}{2}=13.5\ cm^2[/tex]
The Surface Area:
[tex]S.A.=A_\square+4A_\triangle\\\\S.A.=9+4(13.5)=9+54=63\ cm^2[/tex]
There are two numbers whose sum is 53. Three times the smaller number is equal to 19 more than the larger number. What are the numbers?
18 and 35. The numbers whose sum 53 are 18 and 35.
The key to solve this problem is using a system of equations.
There are two numbers whose sum is 53. This number can be represented as x and y. So:
x + y = 53
Three times the smaller number is equal to 19 more than the larger. Let's set x as the smaller number and y the larger number. So:
3x = 19 + y
Clear y in both equations and let's use the equalization method to solve for x:
y = 53 - x and y = 3x - 19
Then,
53 - x = 3x - 19
53 + 19 = 3x + x ---------> 3x + x = 53 + 19 -------> 4x = 72
x = 72/4 = 18
To find y, let's substitute x = 18 in the equation x + y = 53
18 + y = 53 --------> y = 53 - 18
y = 35
X - 9y = -45
Find the x and y intercepts
Answer:
The answer is x = -45 and y = 5
X is -45 and y is 5
What is the value of b in the equation (y^b)^4=1/y^24?
A.) -20
B.)-6
C)6
D.)20
Answer:
b = 6
Step-by-step explanation:
In (y^b)^4=1/y^24 we can re-write (y^b)^4 as y^(4b) and then equate this result to y^24. Setting the exponents equal to one another, we get 4b = 24, or b = 6.
ANSWER ASAP !! TIMED!! Which property is shown in the matrix addition below?
associative property
identity property
inverse property
commutative property
Answer:
yes inverse property
Step-by-step explanation:
Because they all are inverse to each other or oposite
someone help me out. i could use it please. O is inscribed in ABC , which has a perimeter of 76 cal. What is the length of CE?
Answer:
C.E=18 cm
Step-by-step explanation:
we know that
If the circle O is inscribed in triangle ABC
then
A.F=A.D=6 cm
B.D=B.E=14 cm
C.F=C.E=?
The perimeter of triangle is equal to
P=2(6)+2(14)+2(C.E)
P=76 cm
so
76=2(6)+2(14)+2(C.E) -----> solve for C.E
76=12+28+2C.E
2C.E=76-40
C.E=36/2
C.E=18 cm
Consider triangle PQR. What is the length of side QR?
A. 8 units
B. 8/3 units
C. 16 units
D. 16/3 units
ANSWER
C) 16
EXPLANATION
Using the Pythagoras Theorem, we obtain:
QR² =PR²+ PQ²
From the diagram,
[tex]PQ = 8 \sqrt{3} [/tex]
[tex]PR=8[/tex]
We substitute into the formula to get;
[tex]|QR| ^{2} = {8}^{2} + {(8 \sqrt{3} )}^{2} [/tex]
[tex]|QR| ^{2} = 64+ 192[/tex]
[tex]|QR| ^{2} = 256[/tex]
Take square root
[tex]|QR| = \sqrt{256} [/tex]
[tex]|QR| = 16[/tex]
Answer:
The length of side QR is 16 units.
Option C is correct.
Step-by-step explanation:
Given a right angled triangle QPR in which length of sides are
[tex]PQ=8\sqrt3 units[/tex]
[tex]PR=8 units[/tex]
we have to find the length of side QR
As QPR is right angled triangle therefore we apply Pythagoras theorem
[tex](hypotenuse)^2=(Base)^2+(Perpendicular)^2[/tex]
[tex]QR^2=PQ^2+PR^2[/tex]
[tex]QR^2=(8\sqrt3)^2+8^2[/tex]
[tex]QR^2=192+64=256[/tex]
Take square root on both sides
[tex]QR=16 units[/tex]
Hence, the length of side QR is 16 units.
Option C is correct.
How does multiplying a vector by a scalar value of -pi / 4 change the vector?
The answer is:
The second option:
The vector will change direction and decrease in magnitude.
Why?Multiplying a vector by a scalar will modify the magnitude of the vector, depending if the scalar is less or greater than "1" also, when a scalar has a negative sign, the direction of the new vector will be the opposite that the original vector.
Solving
We are given that the scalar value is equal to:
[tex]-\frac{\pi }{4}=-0.79[/tex]
Let's use as original vector, the following vector:
[tex]A=(2,2)[/tex]
Calculating its magnitude, we have:
[tex]|A|=\sqrt{2^{2} +2^{2} }=2.83[/tex]
Then, multiplying the vector by the given scalar, we have:
[tex](-0.79A)=((-0.79)*2,-(0.79)*2)=(-1.58,-1.58)[/tex]
As we can see, the vector changed its direction, since both components are negative.
Calculating the magnitude of the new vector, we have:
[tex]|A'|=\sqrt{(-1.58)^{2} +(-1.58)^{2} }=2.23[/tex]
We can see that the given scalar is less than "1", so the magnitude will decrease, also, the direction is the opposite of the original vector direction since both components have changed its sign.
Hence, we have that the magnitude of the new vector is less than the magnitude of the original vector, also, we can see that the direction of the new vector is the opposite of the original vector direction, so, the answer is the second option, the vector will change direction and decrease in magnitude.
Have a nice day!
Write a rule for the nth term of the sequence 2,20,50,92?
Answer:
482
Step-by-step explanation:
Their 2nd difference is 12
[tex]2,20,50,92,...\\ \\ a_1 = 2\\ a_2 = a_1+18=a_1+6\cdot 3 \\ a_3 = a_2+30=a_2+6\cdot 5\\ a_4 = a_3 +42 = a_3+6\cdot 7\\ a_5 =a_4+54 = a_4+6\cdot 9\\...\\a_n = a_{n-1}+6\cdot (2n-1) \\ \\ a_1+a_2+...+a_n = \\ =a_1+a_2+...+a_{n-1}+2+6\cdot \Big(3+5+7+...+(2n-1)\Big)\\ \\ a_n = 2+6\cdot \Big(3+5+7+...+(2n-1)\Big)\\ a_n = 6\cdot \Big(1+3+5+7+...+(2n-1)\Big)-4 \\ a_n = 6\cdot n^2-4 \\ \\ \Rightarrow \boxed{a_n = 6n^2-4}[/tex]
Two families go to the cinema.
The Smith family buy tickets for one adult and four children
and pay £19
The Jones family buy tickets for two adults and two children
and pay £17
What is the cost of one child’s ticket?
Answer:
Step-by-step explanation:
x= cost of 1 adult ticket
y=cost of 1 kid ticket
so 1 adult+4 childeren=x+4y=19
2 adults and 2 childern=2x+2y=17
elimination
x+4y=19
2x+2y=17
multiply firest equation by -2
-2x-8y=-38
add to first equatuion
-2x-8y=-38
2x+2y=17 +
0x-6y=21
-6y=-21
multiply -2
6y=21
divide by 6
y=21/6=7/2
subsitute
2x+2y=17
2x+2(7/2)=17
2x+14/2=17
2x+7=17
subtract 7 from both sides
2x=10
divide by 2
x=5
1 adult ticket costs £5
1 kid ticket costs £7/2 or £3 and 1/2 or £3.5
hope this helps and give me brainliest pls
in one month a store had 1222 in sales but expenses were 2345 how much money did the store lose in that month
Answer:
$1,123
Step-by-step explanation:
Remember that the profit or loses of a business is given by the subtraction between the revenue and the expenses. In our case the revenue is sales, so:
[tex]profit=sales-expenses[/tex]
[tex]profit=1222-2345[/tex]
[tex]profit=-1123[/tex]
Since the profit is negatives, the business is losing money.
We can conclude that the store lose $1,123 that month
The store lost $1,123 in the given month, which is calculated by subtracting their total expenses of $2,345 from their total sales of $1,222.
To calculate the amount of money the store lost in a month, we need to subtract the expenses from the sales. The formula to find the loss or profit is Profit (or Loss) = Sales - Expenses.
In this scenario:
Sales for the month are $1,222.Expenses for the month are $2,345.Subtract the expenses from the sales: Profit (or Loss) = $1,222 - $2,345.When you perform the subtraction: $1,222 - $2,345 = - $1,123.
The store therefore experienced a loss of $1,123 for that month.