A boat travels about 6 kilometers per hour in still water. If the boat is on a river that flows at a constant speed of
kilometers per hour, it can travel at a speed of (6+r) kilometers per hour downstream. On one particular river, a
boat travels 12 kilometers down stream. The amount of time it takes the boat travel depends on the speed of river
and is expressed by the equation t(r) =
12
(6+r)
The graph is shown below.
What does t(3) mean in this situation?
At what value of r does the graph have a vertical asymptote? Explain
how you know and what this asymptote means in this situation.

The value of x from the inequality equation | 4x + 1 | ≤ 6 , is

-7/4 ≤ x ≤ 5/4

What is an Inequality Equation?

Inequalities are the mathematical expressions in which both sides are not equal. In inequality, unlike in equations, we compare two values. The equal sign in between is replaced by less than (or less than or equal to), greater than (or greater than or equal to), or not equal to sign.

In an inequality, the two expressions are not necessarily equal which is indicated by the symbols: >, <, ≤ or ≥.

Given data ,

Let the equation be A

The value of the equation A is | 4x + 1 | ≤ 6   be equation (1)

From the modulus function rule , we know

If , | x | ≤ a , then -a ≤ x ≤ a

So , on simplifying the equation , we get

-6 ≤ 4x + 1 ≤ 6

Now , the two values are

4x + 1 ≥ -6   be equation (2)

4x + 1 ≤ 6    be equation (3)

From equation (2) , 4x + 1 ≥ -6

Subtracting 1 on both sides , we get

4x ≥ -7

Divide by 4 , we get

x ≥ -7/4

And , From equation (3) , 4x + 1 ≤ 6

Subtracting 1 on both sides , we get

4x ≤ 5

Divide by 4 on both sides , we get

x ≤ 5/4

Therefore , the solution is -7/4 ≤ x ≤ 5/4 and the graph is given below

Hence , The value of x from the inequality equation | 4x + 1 | ≤ 6 , is

-7/4 ≤ x ≤ 5/4

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The mean lifetime of a bulb is 149.37 hours.

What is the meaning of mean?

By dividing the sum of the given numbers by the entire number of numbers, the mean—the average of the given numbers—is determined. Mean is equal to (Sum of All Observations/Total Observations).

The given table represents a continuous data.

The table attached in below. To get mean lifetime of a bulb, find the mean of the given table.

The formula for mean is = Σ fm/N

Where f is frequency and m is midpoint of the class.

N is total number of observation.

From the table, Σ fm = 53175

Total number of bulb is 356.

The mean lifetime is 53175/ 356 = 149.37 hours

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Answer:

15 cats

Step-by-step explanation:

c:d

3:5

5=25

25/5=5

5*3=15

15 cats

======================================================

Explanation:

The tape diagram is shown below.

"C" represents a sub-group of cats, and "D" represents a sub-group of dogs. Each sub-group has the same number of animals.

There are 3 copies of "C" and 5 copies of "D" to represent the ratio 3:5 of cats to dogs. For every 3 cats, there are 5 dogs.

We are told there are 25 dogs available for adoption. This consists of all five blocks of "D" combined. If each "D" is the same number of dogs, then there must be 25/5 = 5 dogs per "D" block. By extension, there are 5 cats per "C" block.

In short, each block has 5 animals.

3 blocks of C = 3*5 = 15 cats are available.

-----------------

Another approach:

C = number of cats available = unknown

D = number of dogs available = 25

C/D = 3/5

C/25 = 3/5

5C = 25*3 ... cross multiply

5C = 75

C = 75/5

C = 15 cats are available

-----------------

A third approach:

The ratio 3:5 scales up to 15:25 after multiplying both parts by 5. This is so the "5" in "3:5" becomes "25".

The ratio of cats to dogs of 15:25 reduces to 3:5 when you divide both parts by the GCF 5.

So this is another way to see how 15 cats are available.

Answer: 12cm^2

Step-by-step explanation: We use the equation 1/2(bh) to get the area. If you put the number in the equation you will get the answer 12.

Answer:

3 pounds:

3 ÷ [tex]\frac{1}{8}[/tex] = 24

7 pounds:

7÷[tex]\frac{1}{8}[/tex] = 56

Step-by-step explanation:

3÷[tex]\frac{1}{8}[/tex] = [tex]\frac{3}{1}[/tex] x [tex]\frac{8}{1}[/tex] = [tex]\frac{24}{1}[/tex] = 24

If it takes 1/8 of a pound to make 1 hamburger.  Then it would take 1 pound to make 8 hamburgers.  

3 x 8 = 24

7x8 = 56

zeroes are 2,-2,

vertical asymptotes are -1,3

plotted on the number line

---------|---------|---------|---------|---------|---------|

      -2          -1          0          1          2         3

What is vertical asymptote?

A vertical asymptote is a line that appears to correspond with a function's graph but never really touches the curve. When graphing a function, the vertical asymptote of the function is crucial.

The curve is neither touched or crossed by a vertical asymptote, which is a vertical line along which the function becomes unbounded (either y tends to or -). K is NOT present in the domain of the function if x = k is the vertical asymptote of a function y = f(x). Any quantity of vertical asymptotes is possible for a function. i.e., it may have 0 vertical asymptote, 1, 2,..., or a limitless amount. If the y-axis represents the vertical asymptote, we often do not portray it by a dotted line. Instead, we represent a vertical asymptote by a vertical dotted line.

1)

zeros means that the value were g(x) is zero

that means (x-2) and (x+2) = 0

x = 2 ,-2

2)

for vertical asymptote (x+1)and (x-3) = 0

x = -1,3

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The formula for y in terms of x in its simplest form is y=5120/27x⁴.

In mathematics, proportionality denotes a linear relationship between two quantities or variables. Both quantities increase in size when one doubles, and both decrease in size when one drops to a tenth of its original value.

Form an equation for y and t first.    

y∝ t²       y=15  t=9

y=kt²

15=k(9²)

15=k(81)

15=81k

Divide both side by 81

5/27 =k

Equation y=5/27t²

Form an equation for t and x

t∝1/x²      t=8  x=2

t=k/x²

8=k/2²

8=k/4

Multiply both side by 4

32=k

Equation: t=32/x²

Replace t with 32/x² in other equation .

y= 5/27 (32/x²)²

y=5120/27x⁴

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We calculate an expected value when the given data points have various probabilities of occurrence .

The expected value , applies best when we are performing the test or experiment  many times. For example, EV applies well to gambling situations to describe expected results for thousands of gamblers per day, repeated day after day after day. However, the EV does not very accurately predict one particular outcome on one specific test.

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Answer:

12.8

Step-by-step explanation:

You can see that triangle DAF and BED are equal.

So lineAF is lineDE.

6.4*2=12.8.

To maximize the profit 29.6 units must be produced.

What is the maxima or minima?

The maxima and minima of a function collectively referred to as extrema in mathematical analysis, are the highest and lowest values of the function, either on the whole domain or within a specific range.

We have given,

The revenue function is,

R(x)=3x

Cost function,

C(x)=0.05x^2+0.04x+10,

Where

x = number of units produced.

Thus, profit = revenue - cost

P(x) = 3x - (0.05x^2+0.04x+10)

= -0.05x^2 + 2.96x - 10

Differentiating with respect to x,

P'(x) = -0.1x + 2.96

Again differentiating with respect to x,

P''(x) = -0.1

For maxima or minima,

P'(x) = 0,

0 = -0.1x + 2.96

-0.1x = -2.96

x = 2.96/0.1

For x = 29.6,

P''(x) = negative,

Hence, to maximize the profit 29.6 units must be produced.

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The linear correlation between the given two variables be,

4.5y + 5.2x = 168.34

Given, data for a classroom of students who measured each (approximately) in inches.

Femur Length 18.7 14.2 20.5 15.9 16.2 13.1 15.0 12.4 19.0 16.2 21.3

Tibia Length  15.8 21.0 16.2 14.3 12.1 15.8 13.0 18.8 14.3 18.7 13.8

we have to find a linear correlation between the two variables,

linear correlation be,

(y - 15.8)/(x - 18.7) = (15.8 - 21.0)/(18.7 - 14.2)

(y - 15.8)/(x - 18.7) = (- 5.2)/4.5

4.5y - 71.1 = -5.2x + 97.24

4.5y + 5.2x = 168.34

So, the linear correlation between the given two variables be,

4.5y + 5.2x = 168.34

Hence, the linear correlation between the given two variables be,

4.5y + 5.2x = 168.34

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a. [tex]123.1(1.069)^3 \approx 150.4[/tex]

b. [tex]123.1(1.069)^{20} \approx 467.5[/tex]

On simplifying for variable, we get -

G = - 20

A mathematical expression is made up of terms (constants and variables) separated by mathematical operators. A mathematical equation is used to equate two expressions. Equation modelling is the process of writing a mathematical verbal expression in the form of a mathematical expression for correct analysis, observations and results of the given problem.

We have the following equation -

(G-2)/(2-24) = 0

We have -

(G-2)/(2-24) = 0

G - 2 = 2 - 24

G - 2 = - 22

G = - 22 + 2

G = - 20

Therefore, on simplifying for variable, we get -

G = - 20

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Answer:

5cm. 8cm

Step-by-step explanation:

Label all the vertices then use Pythagorean theorem

c² = a² + b²

ΔXYZ will also be an equilateral triangle.

What is equilateral triangle?

An equilateral triangle in geometry is a triangle with equal-length sides on all three sides. In the well-known Euclidean space, an equilateral triangle also is equiangular, meaning that each of its three internal angles is 60 degrees and congruent with the others. It's also referred to it as a regular triangle because it is a regular polygon.

As X,Y,Z are the midpoints of sides of equilateral triangle so when we will join the X,Y,Z we will find that distances XY=YZ=XZ are three are equal because this is proved by similarity of triangles which in itself is a detailed and vast topic

Hence XYZ will also be an equilateral triangle

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In this Trigonometric Functions F(x) = 1 + cos(1/X):  n=3 (12.01, 8.10),  n=4 (11.65,8.50), and n=6 (11.24, 9.12), with their upper and lower value

What is Trigonometric Functions?

Trigonometry uses six fundamental trigonometric operations. Trigonometric ratios describe these operations. The sine function, cosine function, secant function, co-secant function, tangent function, and co-tangent function are the six fundamental trigonometric functions. The ratio of sides of a right-angled triangle is the basis for trigonometric functions and identities. Using trigonometric formulas, the sine, cosine, tangent, secant, and cotangent values are calculated for the perpendicular side, hypotenuse, and base of a right triangle.

F(x) = 1 + cos(1/X)

for n=3

Upper sum = 12.01

Lower sum = 8.10

Δx = (b-a)/n = 2π / 3

for n=4

Upper sum = 11.65

Lower sum = 8.50

Δx = (b-a)/n = π / 2

for n=6

Upper sum = 11.24

Lower sum = 9.12

Δx = (b-a)/n = π / 3

Hence, n=3 (12.01, 8.10),  n=4 (11.65,8.50), and n=6 (11.24, 9.12), with their upper and lower value.

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The exact value of sin(u + v), obtained using trigonometric identities is for sine and cosine is 16/65

Trigonometric identities are equations that involve trigonometric functions that are valid for all values of the input variable.

The possible value of sin(u) obtained from a similar question posted online is; sin(u) = 5/13

Both u and v are in quadrant II

The trigonometric identity of the addition formula for the sine of the sum of two angles, indicates;

sin(u + v) = sin(u)·cos(v) + sin(v)·cos(u)

sin(u) = 5/13

cos(v) = -4/5

sin(v) = √(1 - (cos(v))²)

Therefore;

sin(v) = √(1 - (-4/5)²) = 3/5

cos(u) = √(1 - (sin(v))²)

Therefore;

cos(u) = √(1 - (5/13)²) = 12/13

Therefore;

sin(u + v) = (5/13) × (-4/5) + (3/5) × (12/13) = 16/65

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An equation is formed when two equal expressions. The time the same pump will take to empty a tank containing 108,000 gallons of water is 4.5 hours.

When two equal expressions are added together with the aid of the equal sign "=," an equation is created.

Given that the time it takes a water pump to empty a tank directly correlates with the tank's capacity. Consequently, the relationship can be expressed as,

V ∝ T

V = kT

Substituting the volume and time in the equation,

60,000 gallons = 2.5 hours × k

Solving the equation for k,

k = 60,000 gallons / 2.5 hours

k = 24,000 gallons per hour

Now, the equation for the relationship can be written as,

V = 24,000 gallons per hour × T

Now, the time the same pump will take to empty a tank containing 108,000 gallons of water is,

V = 24,000 gallons per hour × T

108,000 gallons = 24,000 gallons per hour × T

108,000 gallons / 24,000 gallons per hour = T

T = 4.5 hours

Hence, the time the same pump will take to empty a tank containing 108,000 gallons of water is 4.5 hours.

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Answer:

(i)  See below for proof.

(ii)  x = 1.823,   x = -0.823

Step-by-step explanation:

Given equation:

[tex]\dfrac{x+2}{x+1}+\dfrac{3}{x}=3[/tex]

Make the denominators of the fractions the same:

[tex]\implies \dfrac{x+2}{x+1}\cdot \dfrac{x}{x}+\dfrac{3}{x}\cdot \dfrac{x+1}{x+1}=3[/tex]

[tex]\implies \dfrac{x(x+2)}{x(x+1)}+\dfrac{3(x+1)}{x(x+1)}=3[/tex]

[tex]\textsf{Apply the fraction rule} \quad \dfrac{a}{c}+\dfrac{b}{c}=\dfrac{a+b}{c}:[/tex]

[tex]\implies \dfrac{x(x+2)+3(x+1)}{x(x+1)}=3[/tex]

Multiply both sides of the equation by x(x + 1) to eliminate the denominator of the fraction on the LHS:

[tex]\implies x(x+2)+3(x+1)=3x(x+1)[/tex]

Expand the brackets:

[tex]\implies x^2+2x+3x+3=3x^2+3x[/tex]

[tex]\implies x^2+5x+3=3x^2+3x[/tex]

Subtract x² from both sides:

[tex]\implies 5x+3=2x^2+3x[/tex]

Subtract 5x from both sides:

[tex]\implies 3=2x^2-2x[/tex]

Subtract 3 from both sides:

[tex]\implies 0=2x^2-2x-3[/tex]

[tex]\implies 2x^2-2x-3=0[/tex]

[tex]\boxed{\begin{minipage}{3.6 cm}\underline{Quadratic Formula}\\\\$x=\dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$\\\\when $ax^2+bx+c=0$ \\\end{minipage}}[/tex]

To solve the quadratic equation  [tex]2x^2-2x-3=0[/tex],  use the quadratic formula.

Therefore:

Substitute these values into the formula and solve for x:

[tex]\implies x=\dfrac{-(-2) \pm \sqrt{(-2)^2-4(2)(-3)}}{2(2)}[/tex]

[tex]\implies x=\dfrac{2 \pm \sqrt{4+24}}{4}[/tex]

[tex]\implies x=\dfrac{2 \pm \sqrt{28}}{4}[/tex]

[tex]\implies x=\dfrac{2 \pm \sqrt{4 \cdot 7}}{4}[/tex]

[tex]\implies x=\dfrac{2 \pm \sqrt{4} \sqrt{7}}{4}[/tex]

[tex]\implies x=\dfrac{2 \pm 2 \sqrt{7}}{4}[/tex]

[tex]\implies x=\dfrac{1 \pm \sqrt{7}}{2}[/tex]

Therefore, the solutions correct to 3 decimal places are:

[tex]x=1.823,\;\; x= -0.823[/tex]

The time that Diana spent looking at the exhibits is given as follows:

1.582 hours.

As stated in the problem, the total time that Diana spent at the museum is given as follows:

3 hours.

The total time that Diana spent at the museum was divided into two activities, presented as follows:

The times of each activity are given as follows:

Then the time that Diana spent looking at exhibits is obtained as follows:

x + 1.418 = 3

The time is obtained by the subtraction of 3 by 1.418 as follows:

x = 3 - 1.418

x = 1.582 hours.

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The signed number to represent ''A car corporation produced 940 fewer cars this month than last'' is - 940.

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Answer:

18k +37

Step-by-step explanation:

13k + 7 + 5(k + 6)

13k + 7 + 5k + 30

13k + 5k + 7 + 30

18k +37

The graph that shows the equation, y = -5x + 45, that represents the scenario given is shown in the attachment.

Determine the slope (m) and the y-intercept (b), then input the values in the equation, y = mx + b, to write the equation of the line that would be graphed.

Given the information above, we have the following:

Let x = hours, and y = quantity  of gallons of paint left

Slope (m) = -5

Y-intercept (b) = 45

Write the equation of the graph by substituting m = 5 and b = 45 into y = mx + b:

y = -5x + 45

The graph shown below for y = 5x + 45 would be labelled as follows:

x-axis = hours

y-axis = quantity of paint (in gallons)

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The system of equations are

The solution of the equation from the graph is

The system of equations is derived from the statements as interpreted below

If x represents the number of hours lifeguarding and y represents the number of hours tutoring

number of hours lifeguarding = x  

number of hours tutoring = y

she can work no more than 11 total hours

x + y not more than 11 hours

x + y < 11

making $14 per hour lifeguarding and making $25 per hour tutoring and must earn no less than $200

$14x + $25y no less than $200

14x + 25y > 200

The required equations are

14x + 25y > 200

x + y < 11

The graph is attached

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The probability that the number 6 will appear between 150 and 200 times inclusively is 0.926.

In the given question, one thousand independent rolls of a fair die will be made.

We have to compute an approximation to the probability that the number 6 will appear between 150 and 200 times inclusively.

Let n=1000 are independent trails.

The probability of success is constant, p=1/6

The probability of success is not constant,

q=1-1/6 = 5/6

let X be the number of times the die shows a 6.

μ = np

μ = 1000*1/6

μ = 166.7

σ=√npq

σ=√1000*1/6*5/6

σ=√5000/36

σ=11.785

Now we used the correlation continuity for finding the probability that number 6 will appear between 150 and 200 times is

P(150≤X≤200)=P(X≤200)-P(X<150)

P(150≤X≤200)=P(Z≤(200+0.5-166.7)/11.785)-P(Z≤(150-0.5-166.7)/11.785)

P(150≤X≤200)=P(Z≤2.87)-P(Z≤-1.46)

P(150≤X≤200)=0.998-0.072

P(150≤X≤200)=0.926

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The value of h ( 12 ) is 26

What is an Equation?

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

Given data ,

Let the equation be A

Now , the value of equation A is

h ( x ) = 4x - 22   be equation (1)

Now , to find the value of h ( 12 ) , substitute the value of x = 12 in the equation , we get

h ( x ) = 4x - 22  

h ( 12 ) = 4 x 12 - 22

On simplifying the equation , we get

h ( 12 ) = 48 - 26

           = 22

Therefore , the value of h ( x ) when x = 12 is 22

Hence , The value of h ( 12 ) is 26

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The statemet 2, the variable K represent carrying capacity and the variable N represent population size, is correct.

In the given question we have to review the video to determine whether each statement about the following equation is true or false.

The given equation is:

[tex]\frac{dN}{dt}=rN(\frac{K-N}{K})[/tex]

The given equation is used to find the annual growth of population.

In which, dN/dt = the annual increase for the population,

r = the annual growth rate,

N = the population size, and

K = the carrying capacity.

So the statemet two, the variable K represent carrying capacity and the variable N represent population size, is correct.

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We have sufficient evidence to conclude that the over all percentages of 3 categories are not same.

The percentage refers to a specific percentage out of a hundred. It is a fraction with the denominator 100, and the sign for it is 1%.

The Latin word per centum, which means "hundred" or "by the hundred," is the source of the word "percent." The Italian phrase per cento, which means "for a hundred," gradually shrank to become the symbol for "percent," which is now used globally. It finally vanished totally. The word "per" was frequently shortened to "p." The contemporary "%" symbol was created by condensing the "cento" to two circles divided by a horizontal line.

Category    Observed([tex]O_{i}[/tex])          Expected([tex]E_{i}[/tex])              ([tex]O_{i}[/tex]-[tex]E_{i}[/tex])²/[tex]E_{i}[/tex]

A or B           100                           120                          3.333

C                   130                           105                          5.9524

D or F             70                            75                            0.056

Total               300                        300                         9.3417

H₀:over percentages are same at UC Davis and at SSC

H₁:They are not same

Test statistic

X²=∑  ([tex]O_{i}[/tex]-[tex]E_{i}[/tex])²/[tex]E_{i}[/tex]

=9.3417

P-value=P(X²(n-1)>9.3417)

=0.0094

Here we can see that the P-value is very low(<0.05 and <0.01)

So, we reject the null hypothesis H₀.

Therefore, we have sufficient evidence to conclude that the over all percentages of 3 categories are not same.

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The amount of money that Margret needs to deposit to reach this amount is $8.

Expression simply refers to the mathematical statements which have at least two terms which are related by an operator and contain either numbers, variables, or both. Addition, subtraction, multiplication, and division are all possible mathematical operations.

Since Margret has $32.00 in a savings account. Margret wants to have a balance of $40.00 in the account, the amount needed will be:

= $40 - $32

= $8

She'll need $8 more.

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