[tex]{ \qquad\qquad\huge\underline{{\sf Answer}}} [/tex]
Here we go ~
For the given function :
[tex]\qquad \sf \dashrightarrow \: f(x) = \dfrac{7}{x - 10} - 7[/tex]
f(x) is defined for all real values of x, except the value of x for which the denominator becomes 0, because if denominator is 0, the function at that instant will be undefined.
that is : f(x) is not defined when :
[tex]\qquad \sf \dashrightarrow \: x - 10 = 0[/tex]
[tex]\qquad \sf \dashrightarrow \: x = 10[/tex]
Hence, f (x) is defined for all real values of x except 10
It can be represented as :
[tex]\qquad \sf \dashrightarrow \: f(x) : x \in R \sim \{10 \}[/tex]
Suppose we want to choose 6 objects, without replacement, from 12 distinct objects. (a) If the order of the choices matters, how many ways can this be done? (b) If the order of the choices does not matter, how many ways can this be done?
(a) There are 665280 ways of selection if the order of the choices matters.
(b) There are 924 ways of selection if the order of the choices does not matters.
(a) If the order of the choices matters, then we must use the permutation formula that is
[tex]^nP_r = \frac{n!}{(n - r)!}[/tex]
Here n is the total no. of objects and r is chosen objects
We have n = 12
r = 6
After we substitute the value we get
[tex]^{12}P_6 = \frac{12!}{(12 - 6)!}[/tex]
[tex]^{12}P_6 = \frac{12!}{6!}[/tex]
[tex]^{12}P_6[/tex] = 665280
Thus, there are 665280 ways of selection if the order of the choices matters
(b) If the order of the choices does not matter, then we must use the combination formula that is
[tex]^nC_r = \frac{n!}{r!(n - r)!}[/tex]
After we substitute the value we get
[tex]^{12}C_6 = \frac{12!}{6!(12 - 6)!}[/tex]
[tex]^{12}C_6 = \frac{12!}{6! 6!}[/tex]
[tex]^{12}C_6 =[/tex] 12×11×10×9×8×7×6! / 6!×6!
[tex]^{12}C_6 =[/tex] 12×11×10×9×8×7/ 6!
[tex]^{12}C_6 =[/tex] 665280/ 720
[tex]^{12}C_6 =[/tex] 924
Thus, there are 924 ways of selection if the order of the choices does not matters.
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Select correct answer from the drop-down menu. Charlie stands at point while holding the control line attached to a model airplane. The plane travels 120 feet counterclockwise from point to point . About how long is the control line? A circle with center point A. C and B are points on the circle. A line connects AC and AB. The angle of A is 80 degrees. The control line is about feet long.
The 120 feet point to point displacement and the 80° angle formed at the vertex A by the control line, AC = AB, Charlie is holding, gives the length of the control line as about 93 feet.
How can the length of the control line be calculated?The given parameters are;
The distance (displacement) traveled by the plane = 120 feet (point to point)
The points on the circle, flown by the model plane = C and B
The center of the circle = A
The angle formed by AC and AB = 80°
Required;
The length of the control line.
Solution:
The control line in the description of the diagram is the line AC or AB
Therefore;
AC = AB
Taking the points C and B as the point the plane flew through, we have;
BC = 120 feet
Which gives
‹ABC = ‹ACB = (180° - 80°)/2 = 50°
From the law of sines, we have;
BC/(sin(A)) = AC/(sin(‹ABC))Which gives;
120/(sin(80°)) = AC/(sin(50°))
AC = 120×(sin(50°))/(sin(80°)) ≈ 93
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Answer:
The control line is about 86 feet long.
Hope this helps!
Step-by-step explanation:
Consider the point \left(x,y\right)(x,y) in the coordinate plane.
What is the rule for translating the point 5 units down?
1. (x,y−5)
2. (x,y+5)
3. (x+5,y)
4. (x−5,y)
Answer:
The first option
Step-by-step explanation:
What are the coordinates of point A?
Answer:
It's coordinates are (1,-2)
Multiple choice: Which of the following steps below would be a good first step when solving the equation 2x+5=11
Divide both sides by 2
Add 5 to both sides of the equation
Subtract 5 from both sides of the equation
Add 11 to both sides of the equation
Answer:
easy
Step-by-step explanation:
How to solve a system of equations by elimination.
Write both equations in standard form.
Make the coefficients of one variable opposites.
Add the equations resulting from Step 2 to eliminate one variable.
Solve for the remaining variable.
Substitute the solution from Step 4 into one of the original equations.
*100 POINTS!* Answer fast
Answer:
[tex](x-6)^2=-1[/tex]
Step-by-step explanation:
Given equation:
[tex]-6x^2+64x-222=-8x[/tex]
When completing the square, first move the terms in x to the left and the constant to the right of the equation:
[tex]\implies -6x^2+64x-222=-8x[/tex]
[tex]\implies -6x^2+64x-222+222=-8x+222[/tex]
[tex]\implies -6x^2+64x=-8x+222[/tex]
[tex]\implies -6x^2+64x+8x=-8x+222+8x[/tex]
[tex]\implies -6x^2+72x=222[/tex]
Factor out the leading coefficient -6 from the left side, then divide both sides by -6:
[tex]\implies -6(x^2-12x)=222[/tex]
[tex]\implies \dfrac{-6(x^2-12x)}{-6}=\dfrac{222}{-6}[/tex]
[tex]\implies x^2-12x=-37[/tex]
Add the square of half the coefficient of the term in x to both sides, forming a perfect square trinomial on the left side:
[tex]\implies x^2-12x+\left(\dfrac{-12}{2}\right)^2=-37+\left(\dfrac{-12}{2}\right)^2[/tex]
[tex]\implies x^2-12x+\left(-6\right)^2=-37+\left(-6\right)^2[/tex]
[tex]\implies x^2-12x+36=-37+36[/tex]
[tex]\implies x^2-12x+36=-1[/tex]
Factor the perfect square trinomial on the left side:
[tex]\implies (x-6)^2=-1[/tex]
To solve:
[tex]\implies \sqrt{(x-6)^2}=\sqrt{-1}[/tex]
[tex]\implies x-6=\pm i[/tex]
[tex]\implies x-6+6=\pm i+6[/tex]
[tex]\implies x=6 \pm i[/tex]
Therefore, the solutions are:
[tex]x=6+i, \quad x=6-i[/tex]
Michael Wittry has been investing in his Roth IRA retirement account for 21 years. Two years ago, his account was worth $215,613. After losing 13 of its original value, it then gained 12 of its new value back. What is the current value of his Roth IRA
If two years ago, his account was worth $215,613. After losing 13 of its original value, it then gained 12 of its new value back. The current value of his Roth IRA is: $215,613.
Current valueFirst step is to calculate the value loss
Value loss=1/3× original value
Value loss= 1/3× $215,613
Value loss= $71,871
Second step is to calculate the new value
New value=$215,613-$71,871
New value= $143,742
Third step is to calculate the value gained
Value gained= 1/2× new value
Valued gained=1/2× $143,742
Valued gained=$71,871
Fourth step is to calculate the current value
Current value= $143,742+$71,871
Current value=$$215,613
Therefore If two years ago, his account was worth $215,613. After losing 13 of its original value, it then gained 12 of its new value back. The current value of his Roth IRA is: $215,613.
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Carol buys last year's best-selling novel, in hardcover, for $25.50. This is with a 15% discount from the original price. What was the original price of the novel?
Answer:
$30.00
Step-by-step explanation:
$25.50 = y(1 - 0.15)
$25.50 = y(0.85)
($25.50) / (0.85) = y
y = $30
Answer:
$30
Step-by-step explanation:
let x = the original price
x - .15x = 25.50
The original price - .15 of the original price is 25.25
.85x = 25.25 Divide both sides by .85
x = 30
Check:
30 - .15(30) = 25.25
30 - 4.50 = 25.25
25.25 = 25.25
Prepare for Rigid Transformations Think about what you know about geometric figures and parallel lines. Fill in each box. Use words, numbers, and pictures. Show as many ideas as you can.
Step-by-step explanation:
Parallel Lines are lines that do not intercept with eachother. They have the same slope yet have different y-intercepts.
Draw two lines not touching.
Example
[tex]y = 5x + 1[/tex]
[tex]y = 5x + 2[/tex]
No examples
[tex]y = 5x + 2[/tex]
[tex]y = - \frac{1}{5} x + 2[/tex]
Hope this helps!
PLS HELP! 50 points!
Xin has a dimes and y nickels. She has a maximum of 18 coins worth a minimum of $1.10 combined. Solve this system of inequalities graphically and determine one possible solution.
The x and y nickels and a dime. She has up to 18 coins totaling a minimum of $1.10 in value. In the graphically representation we can see the line intersect each other at point (12,6).
Given that,
The x and y nickels and a dime. She has up to 18 coins totaling a minimum of $1.10 in value.
We have to create a graphic representation of this inequality system and choose a potential resolution.
x + y ≤18----->equation(1)
Amount in pennies: 5x+10y≥110------>equation(2)
solves by eq2 - 5×eq(1)
5x+10y - 5(x+y) = 120 - 5×18
5x + 10y -5x-5y = 120 - 90
5y = 30
y=6
Then x=12
Therefore, x=12, y=6 is the point where two lines converge.
In the graphically representation we can see the equation (1) line and equation (2) line intersect each other at point (12,6).
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When the number of miles driven equals 0, why is the cost per month not equal to 0? Why does the graph have a positive slope?
Answer:
I think it's because of the down payment and the positive slope because you are going to pay in installments by months.
Shinya went to the salon. The cost if her haircut was $18.00. He tipped the barber 20%. What was the total cost
?
A student writes an incorrect step while checking if the sum of the measures of the two remote interior angles of triangle ABC below is equal to the measure of the exterior angle.
A triangle ABC is shown. The base of triangle extends into a straight line. The angle formed between this straight line and the edge of triangle is marked as d. The angle adjacent to w is marked as c, and the other two angles inside the triangle are marked as a and b.
Step 1: m∠a + m∠b + m∠c = 90 degrees (complementary angles)
Step 2: m∠d + m∠c = 180 degrees (supplementary angles)
Step 3: Therefore, m∠a + m∠b + m∠c = m∠d + m∠c
Step 4: So, m∠a + m∠b = m∠d
In which step did the student first make a mistake and how can it be corrected? (1 point)
a
Step 1; it should be m∠a + m∠b + m∠c = 180 degrees (corresponding angles)
b
Step 1; it should be m∠a + m∠b + m∠c = 180 degrees (sum of angles of a triangle)
c
Step 2; it should be m∠d − m∠c = 90 degrees (alternate interior angles)
d
Step 2; it should be m∠d − m∠c = 90 degrees (adjacent angles)
Answer:
B
Step-by-step explanation:
from seeing the steps listed the answer should be b would be helpful if u posted the diagram along
write a explicit formula 96 24 6
Answer:
[tex]a_{n}[/tex] = 96 [tex](\frac{1}{4}) ^{n-1}[/tex]
Step-by-step explanation:
there is a common ratio between consecutive terms, that is
[tex]\frac{24}{96}[/tex] = [tex]\frac{6}{24}[/tex] = [tex]\frac{1}{4}[/tex]
this indicates the sequence is geometric with explicit formula
[tex]a_{n}[/tex] = a₁ [tex](r)^{n-1}[/tex]
where a₁ is the first term and r the common ratio
here a₁ = 96 and r = [tex]\frac{1}{4}[/tex] , then explicit formula is
[tex]a_{n}[/tex] = 96 [tex](\frac{1}{4}) ^{n-1}[/tex]
How do we add/subtract fractions and mixed fractions?
The addition and subtraction of fractions and mixed fractions can be done by finding the lowest common multiple (LCM) of the denominators before simplifying.
Addition and subtraction of fractionGiven 3 1/2 + 2 3/4
= 7/2 + 11/4
Find the LCM of both denominators= (14+11) / 4
= 25/4
= 6 1/4
Given 3 1/2 - 2 3/4
= 7/2 - 11/4
Find the LCM of both denominators
= (14-11) / 4
= 3/4
Therefore, the addition and subtraction of fractions and mixed fractions can be done by finding the lowest common multiple (LCM) of the denominators before simplifying.
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An appliance store decreases the price of a 19-In television set 29% to a sale price of $442.69. What was the original price?
The original price is $623.51.
How to compute the price?Let the original price be represented by x.
Since the appliance store decreases the price of a 19-In television set 29%, that means the percentage of the sales price will be:
= 100 - 29
= 71%
Therefore, the original price will be:
= 71% of x = $442.69
0.71x = $442.69
x = $442.69/0.71
x = $623.51
The price is $623.51.
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The price of a McChicken sandwich at McDonald's was $1.00 in 2018, and the sandwich
is now $2.44 in 2022. Find the percent increase of the sandwich that occurred from
2018 to 2022. (Round to the nearest whole percent)
Gabby worked 30 hours in 4 days. Determine the rate for ratio of the two different quantities.
30/34 hours per day
4/30 hours per day
30/4 hours per day
4/34 hours per day
The required rate for ratio of the two different quantities i.e
Gabby worked 30/4 hours per day.
Option (c) is correct.
What is arithmetic?Arithmetic is the branch of mathematics that deals with the study of numbers using various operations on them. Basic operations of math are addition, subtraction, multiplication and division. These operations are denoted by the given symbols.
Given:
Gabby worked 30 hours in 4 days.
According to given question we have
The ratio of the two quantities of the different kind and in the different units is a fraction that shows how many times one quantity is of the other.
By the use of arithmetic we have,
Gabby worked 30 hours in 4 days means
4 days= 30 hours
1 days = 30/4 hours
Therefore, the required rate for ratio of the two different quantities i.e
Gabby worked 30/4 hours per day.
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Answer: 30/4 hours per day
Step-by-step explanation:
Stephan monthly pay stub indicates that his monthly gross income is 3800 however 800 is withheld for income and social security taxes and 200 is withheld for his health and disability insurance how much is Stephan’s Disposable income
Step-by-step explanation:
3800 is the general amount, but before he can touch it, 800 and 200 are taken away from it for the described purposes.
so, what is left is
3800 - 800 - 200 = 2800
this is then the real amount he can do something with.
A triangle has one 30° angle, an unknown angle, and an angle with a measure that is twice the measure of the unknown angle. Find the measures of the triangle's unknown angles and explain how you found the answer.
The measures of the triangle's unknown angles will be 50° and 100°.
What is the triangle?A triangle is a three-sided polygon with three angles. The angles of the triangle add up to 180 degrees.
A triangle has one 30° angle, an unknown angle, and an angle with a measure that is twice the measure of the unknown angle.
Let 'a' and 'b' be the unknown angles of the triangle. Then the equation will be
∠a = 2 ∠b
We know that the angle sum of the triangle is 180°. Then the equation will be
∠a + ∠b + 30° = 180°
2∠b + ∠b = 150°
3∠b = 150°
∠b = 50°
Then the measure of the angle 'a' will be
∠a = 2 ∠b
∠a = 2 x 50°
∠a = 100°
The measures of the triangle's unknown angles will be 50° and 100°.
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(100 POINTS!) Evaluate each expression given that the variables have the following values: (image below)
A town's population has been growing linearly. In 2003, the population was 59,000
and the population has been growing by 1,700 people each year. Let P represent the
population, and the number of years after 2003.
Answer
59,000+1,700=60,000
Step-by-step explanation:
the day of the month
a) interval
b)ordinal
c) nominal
d) ratio
?
The day of the month is nominal (option c).
What is nominal data?Nominal data is data that can be labelled or classified into mutually exclusive categories within a variable. A nominal variable is a categorical variable that is qualitative.
A ratio scale is quantitative. A ratio scale is a variable with a true zero. An example is weight. An interval variable is a variable where each value is measured along a scale. An example of an interval variable is credit score.
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QUESTION 5
(a) Differentiate the following
(i)
y=e²x - 3e-4x
Answer: e² - 4
Step-by-step explanation:
Differentiate of e²x - 3e-4x is e² - 4
Answer:
[tex]\dfrac{\text{d}y}{\text{d}x} = 2e^{2x}+12e^{-4x}[/tex]
Step-by-step explanation:
Given function:
[tex]y=e^{2x}-3e^{-4x}[/tex]
[tex]\boxed{\begin{minipage}{3.7cm}\underline{Differentiating $ax$}\\\\If $y=ax$, then $\dfrac{\text{d}y}{\text{d}x}=a$\\\end{minipage}}[/tex]
[tex]\boxed{\begin{minipage}{5.5 cm}\underline{Differentiating $e^{f(x)}$}\\\\If $y=e^{f(x)}$, then $\dfrac{\text{d}y}{\text{d}x}=f\:'(x)e^{f(x)}$\\\end{minipage}}[/tex]
Therefore:
[tex]\begin{aligned}\implies \dfrac{\text{d}y}{\text{d}x} & = \dfrac{\text{d}}{\text{d}x} e^{2x}-\dfrac{\text{d}}{\text{d}x}3e^{-4x}\\\\& =2 \cdot e^{2x}-(-4)\cdot 3 e^{-4x} \\\\ &= 2e^{2x}+12e^{-4x}\end{aligned}[/tex]
Hannah owns a food truck that sells tacos and burritos.She sells each tacos for $3.75 and each burritos for$7.75 . Hannah must sell no less than $430 worth of tacos and burritos each day
given z (x)=15x-x^2+3
Answer:
○ [tex]z(x - 1) = -x^2 + 17x - 13[/tex]
Step-by-step explanation:
We are told that:
[tex]z(x)= 15x - x^2 + 3[/tex],
and told to find an expression for [tex]z(x - 1)[/tex].
In order to find [tex]z(x - 1)[/tex], we have to replace [tex]x[/tex] in the definition of [tex]z(x)[/tex] with [tex](x-1)[/tex]:
[tex]z(x - 1) = 15(x -1) - (x - 1)^2 + 3[/tex]
Now we can simplify:
⇒ [tex]z(x-1) = 15x - 15 - (x - 1)^2 + 3[/tex] [Distributing 15 into the first brakets]
⇒ [tex]z(x-1) = 15x - 15 - (x^2 - 2x + 1) + 3[/tex]
⇒ [tex]z(x-1) = 15x - 15 - x^2 + 2x - 1 + 3[/tex] [Distributing the minus sign]
⇒ [tex]z(x - 1) = 17x - x^2 -13[/tex] [Combining like terms]
⇒ [tex]z(x - 1) = -x^2 + 17x - 13[/tex]
Therefore, the first option is the correct one.
calculate the VAT charged on this amount R1599.00(VAT included)
Answer:
VAT = R239.85
Step-by-step explanation:
VAT = 15%
15/100 = 0.15
R1599 × 0.15 = 239.85
VAT = R239.85
A dog is 9 times as heavy as a cat, a hamster is 20 times as light as a cat, and a mouse is 6 times as heavy as the hamster. How many times is the dog as heavy as a mouse?
Answer:
30
Step-by-step explanation:
Let the cat have a weight of x.
Then, the dog weighs 9x.
The hamster weighs x/20.
The mouse weighs 3x/10.
So, the dog is 9/(3/10) = 30 times as heavy.
The weight of the dog is 30 times the weight of the mouse.
What is simplification?The process in mathematics to operate and interpret the function to make the function or expression simple or more understandable is called simplifying and the process is called simplification.
Let the weight of the dog be x,
A dog is 9 times as heavy as a cat,
weight of the dog = 9x - - - - -(1)
A hamster is 20 times as light as a cat
Weight of the hamster = x /20
A mouse is 6 times as heavy as a hamster.
Weight of the mouse = 6(x /20)
m(20/6) = x
Now, put x = m(20/6) in equaiton 1,
weight of the dog = 9[m(20/6)]
weight of the dog = 30m
Thus, the weight of the dog is 30 times the weight of the mouse.
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alan invests £1000 in a bank account with an interest rate of r% per year. after one year the value of the account is £1025.
calculate r.
Bank account with an interest rate of r% per year = 2.5%
Simple Interest :
Simple interest paid or received over a certain period is a fixed percentage of the principal amount.
Simple Interest=P×[tex]\frac{r}{100} }[/tex]×T
where:
P=Principal
r= interest rate on a certain period
T= Time
Given :
Principal(P)= £1000
Time (T)= 1 Year
Find :
the interest rate per year = r%
Now,
Interest earned on account = £1025 - £1000
Simple Interest= £ 25
Simple Interest=P×[tex]\frac{r}{100}[/tex]×T
25= 1000×[tex]\frac{r}{100}[/tex]×1
25×100=1000r
2500=1000r
[tex]\frac{2500}{1000} =r\\\frac{25}{10} =r\\\frac{5}{2} =r\\2.5=r[/tex]
r= 2.5%
the interest rate of r% per year. after one year the value of the account=2.5%
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A city has a population of 210,000 people. Suppose that each year the population grows by 4.5% . What will the population be after 5 years?
The population of a city after 5 years would be 261698.
We know the the formula for the exponential growth of the population is,
P(t) = a(1+r)^t
where, a is the initial population
t is the time period
r is the rate of growth
In this question, we have a = 210000, %R = 4.5
So, r = 0.045
We need to find the population after 5 years.
t = 5
Using above formula,
P = 210000 × ( 1 + 0.045)^5
P = 210,000 × (1.045)^5
P = 210,000 × 1.25
P = 261698.2
P ≈ 261698
Therefore, the population of a city after 5 years would be 261698.
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