If y varies directly as x, find the constant of variation and the direct variation equation for the situation.
y = 0.1 when x = 0.2

[tex]x=2.16*10^{11}u-7.2k[/tex].

Side note: This answer could be more precise if made usage of more significant figures. For practicality reasons, in this solution we reduced the amount of significant figures to use.

[tex]150u=0.694(x+7.2K)*10^-9[/tex]

[tex]150u=[(0.694*x)+(0.694*7.2K)]*10^-9\\\\150u=[0.694x+4.9968k]*10^-9\\ \\150u=10^-9*0.694x+10^-9*4.9968k[/tex]

[tex]150u-10^-9*0.694x=10^-9*0.694x-10^-9*0.694x+10^-9*4.9968k\\ \\150u-10^-9*0.694x=10^-9*4.9968k[/tex]

[tex]-150u+150u-10^-9*0.694x=10^-9*4.9968k-150u\\ \\-10^-9*0.694x=10^-9*4.9968k-150u[/tex]

5. Divide both sides by "-10^-9*0.694".

[tex]\frac{-10^-9*0.694x}{-10^-9*0.694} =\frac{10^-9*4.9968k-150u}{-10^-9*0.694} \\ \\\frac{-10^-9*0.694x}{-10^-9*0.694} =\frac{10^-9*4.9968k}{-10^-9*0.694}-\frac{150u}{-10^-9*0.694} \\ \\x=-7.2k-(-2.16*10^{11} u)\\ \\x=-7.2k+2.16*10^{11}u\\ \\x=2.16*10^{11}u-7.2k[/tex]

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For the given geometric sequence the seventh term is equal to 19/81 and the 12th term is equal to "-57" the value of the 15th term is 1539.

What is a geometric sequence?

One particular sort of sequence is a geometric sequence. It is a sequence in which each term, with the exception of the first, is multiplied by a fixed integer to obtain the following term. The common ratio must be multiplied by the current term in the geometric sequence to obtain the next term, and the common ratio must be divided by the current term to obtain the previous term.

Solution Explained:

We know, a7 = 19/81, a12 = -57

So, ar^7 = 19/81 and ar^12 = -57     (r= common ratio)

Since r< a15 < 0 [the sign alternate]

Ignoring the signs,
r^12 = r^7 X r^5
57 = 19/81 X r^5

(57 X 81)/19 = r^5

243 = r^5

r = the fifth ratio of 243 = 3

therefore, r = -3

So, a15 = a12 X r^3
            =  -57 X (-3)^3
            = 1539

Hence, the value of the 15th term is 1539.

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Since the function is not one-to-one, the inverse function of f(x) = x³ + 3x² - 4 is not defined at x = -4.

An original function is composed by a set of points in the following format:

(x,y).

On the graph of the function, we have that:

On the inverse function, the input and the output are exchanged, hence the format of the point is:

(y,x).

To find the numeric value of the inverse function at x = -4, we need to identify the value of x on the original function where y = -4.

However, looking at the graph of the function given at the end of the answer, the function is not one-to-one at y = -4, since f(-3) = f(0) = 4, thus the inverse function is not defined at x = -4.

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When only 17 posters have been printed, it will cost $ 5. 46to print 143 more.

A mathematical formula known as a cost function is used to visualise how manufacturing costs will vary depending on the output level. In other words, given a specified quantity produced, it calculates the entire cost of production.

The following equation, C'(x)=x-3/4, tells us the marginal cost (in dollars) of printing a poster after x posters have been printed.

The given equation is :C'(x)=x-3/4

The cost of printing 133 more posters when 17 have already been printed is given by;

applying the limitations we obtain and integrating both sides of the equation;

[tex]$$\int_a^b C^{\prime}(x) d x=\int_{17}^{133} x^{\frac{-3}{4}} d x$$[/tex]

As we know that [tex]$\int x^n d x=\frac{x^{n+1}}{n+1}$[/tex], so;

[tex]$&=4\left[x^{\frac{1}{4}}\right]_{17}^{133} \\[/tex]

[tex]$&=4\left[(133)^{\frac{1}{4}}-(17)^{\frac{1}{4}}\right] \\[/tex]

= $ 5.4616

Therefore When only 17 posters have been printed, it will cost $ 5. 46to print 143 more.

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The value of sin 3° will be;

⇒ sin 3° = 1/8 (√(3 - √5) (√3 + 1) - √ (5 + √5) (√3- 1)

What is an expression?

Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.

Given that;

The values are,

sin 18° = √(3 - √5) / 2√2

cos 18° = √ (5 + √5) / 2√2

Now,

Find the value of sin 3° as;

⇒ sin 3° = sin (18 - 15)°

   = sin 18° cos 15° - cos 18° sin 15°

   = √(3 - √5) / 2√2 × (√3 + 1) / 2√2  - √(5 + √5) / 2√2 × (√3-1)/2√2

   = 1/8 (√(3 - √5) (√3 + 1) - √ (5 + √5) (√3- 1)

Thus, The value of sin 3° will be;

⇒ sin 3° = 1/8 (√(3 - √5) (√3 + 1) - √ (5 + √5) (√3- 1)

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Answer:

48 square inches

Step-by-step explanation:

A=1/2 bh

A=(1/2)(8)(12)

A=(4)(12)

A=48 square inches

you can also multiply 8 x 12 then divide by 1/2 or 2 which is still 48. either way is correct but since the formula is half of the base times height that is the step i followed. Also area is always whatever measurement used (in this case inches) squared.

Answer:

The last number is 4

Step-by-step explanation:

Answer:

Step-by-step explanation:

Each exponent on the left side is a positive number and therefore their sum is positive too.

But we have a negative value on the right.

This is not possible, hence no solutions.

We're assuming Deshaun is driving at a constant speed/at the same rate,

He drove 469 miles in 7 hours -> his speed = 469 : 7 = 67 (mph)

Therefore, it would take him 335 : 67 = 5 hours to drive 335 miles.

Answer:

Step-by-step explanation: 348 miles and 6 hours

The answer will be the graph of the straight line.

What is the equation straight line?

Y = mx + c is the general equation for a straight line, where m denotes the line's slope and c the y-intercept. It is the version of the equation for a straight line that is used most frequently in geometry. There are numerous ways to express the equation of a straight line, including point-slope form, slope-intercept form, general form, standard form, etc. A straight line is a geometric object with two dimensions and infinite lengths at both ends. The formulas for the equation of a straight line that are most frequently employed are y = mx + c and axe + by = c. Other versions include point-slope, slope-intercept, standard, general, and others.

Using the equation we can plot the graph in 2D.

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The value of v from the given equation v=u+at is 42

An equation is a mathematical expression that contains an equals symbol

From the given equation v=u+at

u=18

a=6

t=4

Substituting for u,a and t in the equation

v=18+6(4)

expanding the bracket

v=18+24

v=42

Hence, the the v in the equation v=u+at is 42

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The composite function ( f. g(x)) is x² + 4x + 3

A function can be described mathematically as a rule, law or equation that explains, shows and elaborates the relationship between to variables.

The variables are known as;

From the information given, we have the functions as;

To determine the composite function ( f. g(x)), we have to substitute the value of g(x) as a function in the function f(x) as the variable 'x', we get;

( f. g(x)) = (x + 2)² - 1

Now, let's expand the square

( f. g(x)) = (x + 2) ( x+ 2) - 1

Expand the bracket

( f. g(x)) = x² + 2x + 2x + 4 - 1

Now, let's collect like terms, we get

( f. g(x)) = x² + 4x + 3

Thus, the function is x² + 4x + 3

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The complete question:

Find ( f. g(x)) if f(x)=x^2-1, g(x)=x+2

The co-ordinates of the mid-point of the line PQ are (6,4).

From the question, we have

The point P has co-ordinates (10, 12) and the point Q has co-ordinates (2,-4).

Mid-point of the line PQ =(10+2)/2 , (12-4)/2

= (6,4).

Midpoint:

A location in the middle of a line connecting two points is referred to as the midpoint. The midpoint of a line is located between the two reference points, which are its endpoints. The line that connects these two places is split in half equally at the halfway. In addition, the halfway is reached if a line is drawn to divide the line that connects these two places. When we know the coordinates of two points, we can apply the midpoint formula to get the midpoint between them. If we know the coordinates of the other endpoint and the midpoint, we can use the midpoint formula to obtain the coordinates of the endpoint as well.

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Answer:

Step-by-step explanation:

9.65+0.4=10.05

16.058-1.2=14.858

Answer:

a.  x = 1.75,  x = 4.25

b.  y = -7

c.  Maximum point at (3, 5).

d.  Domain:  (-∞, ∞)

e.  Range:  (-∞, 5]

f.  See attachment.

Step-by-step explanation:

Given function:

[tex]\text{f}(x)=-4|x-3|+5[/tex]

As per the question, use a graphing calculator to graph the given function.

The x-intercept(s) are the points at which the curve crosses the x-axis.

The y-intercept is the point at which the curve crosses the y-axis.

From inspection of the graph:

The domain of a function is the set of all possible input values (x-values).

The domain of the given function is unrestricted.

The range of a function is the set of all possible output values (y-values).

As the function has a maximum point at y = 5, the range is restricted.

Graph label the axes using a scale of 1 (see attachment).

Substitute x = 5 into the function to find the value of y at that point:

[tex]\begin{aligned}\implies \text{f}(5)&=-4|5-3|+5\\&=-4|2|+5\\&=-4(2)+5\\&=-8+5\\&=-3\end{aligned}[/tex]

When the students get reduced to 15 then each student will get 3 pens.

The unitary technique involves first determining the value of a single unit, followed by the value of the necessary number of units.

For example, Let's say Ram spends 36 Rs. for a dozen (12) bananas.

12 bananas will set you back 36 Rs. 1 banana costs 36 x 12 = 3 Rupees.

As a result, one banana costs three rupees. Let's say we need to calculate the price of 15 bananas.

This may be done as follows: 15 bananas cost 3 rupees each; 15 units cost 45 rupees.

Given:

Total students = 25

Each Student get 2 pens.

So, 25 students get = 25 x 2= 50 pens

Now, number of children reduced = 25 -10= 15

So, each student get = 50/ 15 = 3.333 = 3 pens

Hence, each student will get 3 pens.

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The time that the three buses will next arrive at the station, given the time between their individual arrivals is 10 am.

Service A arrives every 15 minutes. Service B arrives every 20 minutes. Service C arrives every 30 minutes.

The goal is to find the time when all services arrive at the same time. This can be found by finding the LCM of all three times. This will show the shortest time till all services arrive.

The LCM of 15 minutes, 20 minutes, and 30 minutes is 60 minutes.

The three bus services will arrive at the same time in 60 minutes which is:

= 9 am + 60 minutes

= 10 am

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The percent of change each year is 15%.

How to calculate percentage?

A value or ratio that may be stated as a fraction of 100 is referred to as a percentage in mathematics. If we need to calculate a percentage of a number, we should divide it by its entirety and then multiply it by 100. The proportion therefore refers to a component per hundred. Per 100 is what the word percent means. The letter "%" stands for it.

It is given that The number of carbon atoms in a fossil is given by the function y = [tex]51000(0.85)^t[/tex], where t represents the number of years since being discovered.

Here , y  =  [tex]51000(0.85)^t[/tex]

t = number of years

y = number of carbon atoms in a fossil.

For t = 1 year

y = 51000[tex](0.85)^1= 51000(0.85) = 43350[/tex]

For t = 2 years

y = 51000[tex](0.85)^2= 36847.5[/tex]

For t = 3 years

y = 51000[tex](0.85)^3=31320.375[/tex]

Percentage change in one year = [tex]\frac{36847.5-43350}{43350}[/tex] × 100 = -15%

Percentage change in second year =  [tex]\frac{31320.375-36847.5}{36847.5}[/tex] × 100 = -15%

here, negative sign shows decrement. In this way, every year carbon atom in fossil decreases by 15%.

Therefore, the percent of change each year is 15%.

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Answer: 3274 cookies

Step-by-step explanation:

She sold : 2016 ÷ 8 = 252 ( bags )

The remaining cookies : 12 × 252 = 3024 ( cookies )

She baked : 3024 + 250 = 3274 ( cookies )

Using prime factorization, the LCM and GCF of numbers are

1) For 46 and 4

GCF = 2

LCM = 92

2) For 2 and 34

GCF = 2

LCM = 34

3) For 32 and 4

GCF = 4

LCM = 32

A natural number other than 1 with just the number 1 and itself as factors is known as a prime factor. Actually, 2, 3, 5, 7, 11, and so on are the first few prime numbers. For numbers, we may now also employ what is known as prime factorization, which really involves utilizing factor trees.

When a number is expressed as the product of its prime factors, this is known as prime factorization.

1) The factors of the numbers are

46 = 2 × 23

4 = 2 × 2

GCF = 2

LCM = 2 × 2 × 23

= 92

2) The factors of 34 are

34 = 2 × 17

GCF = 2

LCM = 2 × 17

=34

3) The factors of the numbers are

32 = 2 × 2 × 2 × 2 × 2

4 = 2 × 2

GCF = 2 × 2

= 4

LCM = 2 × 2 × 2 × 2 × 2

= 32

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Answer:5) gcf=6 lcm=42 | 6) gcf=2 lcm=270 | 7) gcf=2 lcm=224

Step-by-step explanation:

The pattern is that two larger numbers differ by 2 and the smaller is the root of twice the sum of the larger two.

The given formula gives rise to the above observations. it tells you

a = 2x

b = x^2 -1

c = x^2 +1

This is a special case of Euclid's formula ...

a = 2mn

b = m^2 -n^2

c = m^2 +n^2

for n=1.

Since m and n can be any integer values, there are certainly many other. Pythagorean triples that do not match the formula given in the problem. Using n=2, for example, the triples are:

a = 4x

b = x^2 -4

c = x^2 +4

Some triples from this set of formulas are (5, 12, 13), (20, 21, 29), (28, 45, 53), (36, 77, 85).

It should bm be noted that triples from this set of formulas will not match directly those from the formulas in the problem statement, but triples from both lists may reduce to the same primitive triple.

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Complete question

Examine this set of Pythagorean triples from part C. Look for a pattern that is true for each triple regarding the difference between the three values that make up the triple.

Describe this pattern. Then see if you can think of another Pythagorean triple that doesn’t follow the pattern you just described and that can’t be generated using the identity (x2 − 1)2 + (2x)2 = (x2 + 1)2. Explain your findings.

x-value Pythagorean Triple

3 (6,8,10)

4 (8,15,17)

5 (10,24,26)

6 (12,35,37)

The equation used to determine height and time is y = 16x² + 96x + 10 and y = 17x + 5

What is Projectile Motion ?

A projectile is something that has only one force acting on it the entire time, which is gravity, with the exception of the beginning.There are numerous types of projectiles.A projectile is something that is launched from the position of rest.A projectile is also any object that is hurled vertically upward as long as air resistance has no effect. Additionally, a projectile is any object that is fired upward at an angle to a horizontal plane.

The height of projectile after t seconds is given,

h(t) = 1/2 gt² + v₀t + s₀

where,

v₀ ( initial velocity ) = 96 feet per second

s₀ ( initial height ) = 10 feet

g = 32 feet per second²

put the values,

h(t) = 1/2 (32)t² + 96 t + 10

h(t) = 16 t² + 96 t + 10

so, let h(t) = y and t = x

y = 16x² + 96x + 10

We know, Amy's bullet follows a linear path,

The linear equation is,

y = mx + b

where, slope (m) = 17 feet per second and,

y-intercept (b) = 5 feet

plug in the values in linear equation,

y = 17x + 5

Therefore, The equation used to determine height and time is y = 16x² + 96x + 10 and y = 17x + 5

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The percentages to complete the table for the steaming time are 40 min, 20 min and 100 min respectively

A percentage is a number or ratio that can be expressed as a fraction of 100.

Given: Percentage: 100% of 40 min, 50% of 40 min and 250% of 40 min

100% of 40 min:

100% of 40 min = (100/100) × 40 min = 1 × 40 min = 40 min

50% of 40 min:

50% of 40 min = (50/100) × 40 min = 0.5 × 40 min = 20 min

250% of 40 min:

250% of 40 min = (250/100) × 40 min = 2.5 × 40 min = 100 min

Therefore, the value of the percentages to complete the table for the steaming time are 40 min, 20 min and 100 min for 100% of 40 min, 50% of 40 min and 250% of 40 min respectively.

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