What is the solution to the equation One-fourth x minus one-eighth = Start Fraction 7 Over 8 End Fraction + one-half x?
x = negative 5
x = negative 4
x = 4
x = 5

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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If one serving is 1/4 of the pizza, then the number of servings left is 10.5 servings

Total number of pizza = 3

Each pizza was cut into 8 slice.

Total number of slices = 3 × 8

= 24 slices

He ate 3 slice of pizza

Remaining slices = 24 - 3

= 21 slices

The fraction pizza that left = 21/24

If one serving of pizza is 1/4 of the pizza, that is the 1/12 of three pizza

Number serving left = The fraction pizza that left ÷  one serving of pizza

= 21/24 ÷ 1/12

= 21/24 × 12

= 10.5 servings

Hence, there will 10.5 servings left

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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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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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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 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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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.

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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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 length of the other leg of the given right triangle is [tex]\sqrt{70}[/tex].

A triangle is classified as a right triangle when it presents one of your angles equal to 90º.  The greatest side of a right triangle is called the hypotenuse. And, the other two sides are called legs.

The math tools applied for finding angles or sides in a right triangle are the trigonometric ratios or the Pythagorean Theorem.

The Pythagorean Theorem says (hypotenuse)²= (leg1)²+(leg2)². And the main trigonometric ratios are: sin (x) , cos  (x) and tan  (x) .

The question gives:

Therefore, from the Pythagorean Theorem, you will find the value of the other leg. See below.

[tex]9^2={\sqrt{11}^2 +{leg_2}^2[/tex]

[tex]81=11 +{leg_2}^2[/tex]

[tex]{leg_2}^2=81-11\\ \\ {leg_2}^2=70\\ \\ leg_2=\sqrt{70}[/tex]

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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

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: a

Step-by-step explanation:

The value of n from the given parallel lines is 6. Therefore, the option A is the correct answer.

Angles in parallel lines are angles that are created when two parallel lines are intersected by another line called a transversal.

Given that, the angles from the parallel lines are (3n+16)° and (5n+4)°.

Here, (3n+16)°=(5n+4)° (Interior alternate angles are equal)

5n-3n=16-4

2n=12

n=6

Therefore, the option A is the correct answer.

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The journey along Line segment X Y is longer than the path along Line segment Z Y, and the path along Line segment Z Y is longer than the path along Line segment A B, hence Option-A is the correct answer.

Given that,

He makes paths that follow Line segments A and B, X and Y, and Z and Y through the garden.

We have to find which accurately compares the pathways' lengths.

We know that,

The length of the third side in a triangle is proportional to the angle Ф  if the lengths of the two sides that make up the angle Ф are fixed.

XY > YZ because XY's opposing angle (79°) is greater than YZ's (65°).

As before, YZ > AB because the opposing angle of YZ (65°) is greater than that of AB (36°).

Since both sides of the angle are the same length (8 feet),

As a result, assertion A. is accurate. (Answer)

Therefore, The journey along Line segment X Y is longer than the path along Line segment Z Y, and the path along Line segment Z Y is longer than the path along Line segment A B, hence Option-A is the correct answer.

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Answer:

It's A

Step-by-step explanation:

I got 100 on the quiz

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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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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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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The angles ∠BCD and ∠BAD are opposite interior angles of a parallelogram. in which the alternate interior angles  ∠BAC  and  ∠ACD and ∠BCA and ∠DAC are congruent, from which by angle addition property, ∠BCD ≅ ∠BAD

Alternate interior angles are angles formed by the transversal to two parallel lines on the inside of the parallel lines, and on opposite sides of the transversal.

The two column-method used to  prove that ∠BCA is congruent to ∠DAC can be presented as follows;

Step       [tex]{}[/tex]         Statement                                          Reason

1.        [tex]\overline{BC}[/tex]║ [tex]\overline{AD}[/tex],  [tex]\overline{AB}[/tex]║[tex]\overline{CD}[/tex]   [tex]{}[/tex]       Given

2. [tex]{}[/tex]    ∠BCA ≅ ∠DAC                  ║ lines cut by a trans. form ≅ int.  ∠s

3. [tex]{}[/tex]    [tex]\overline{AC}[/tex]║[tex]\overline{CA}[/tex]     [tex]{}[/tex]                         Reflexive Property

4.     ∠BAC ≅ ∠ACD [tex]{}[/tex]                  Alternate interion angles

5.     ∠BCA = ∠DAC[tex]{}[/tex]                    Definition of congruency

[tex]{}[/tex]        ∠BAC = ∠ACD

6.    ∠BCD = ∠BCA + ∠ACD[tex]{}[/tex]       Angle addition property

      ∠BAD = ∠BAC + ∠DAC

7.    ∠BCD = ∠BCA + ∠ACD  [tex]{}[/tex]       Substitution property

8. [tex]{}[/tex]   ∠BCD = ∠BAD   [tex]{}[/tex]                    Substitution property

9.    ∠BCD ≅ ∠BAD   [tex]{}[/tex]                    Definition of congruency

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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]

a) k = 1/2 then v = 1 cubic unit

b) k = 0.6 then v = 1.728 cubic units

c) k = 1 then v = 8 cubic units

d) k = 1.5 then v = 27 cubic units

The value of scale factor k, which is used to obtain a solid with volume v cubic units by 8 cubic unit is, ∛v / 2.

Here the new volume of the solid is find out by the volume of 8 units and the scale factor k. Thus the scale factor k should be equal to the ratio of the volume of these two solid.

The volume of the solid is represents in cubic units. Thus, k = ∛(v/2)

k = (∛v) / 2

a) k = 1/2

(1/2)³ * 2³ = v

v = 1 cubic units

b) k = 0.6

(0.6)³ * 8 = v

0.216 * 8 = v

v = 1.728 cubic units

c) k = 1

1³ * 8 = v

v = 8 cubic units

d) k = 1.5

(1.5)³ * 8 = v

3.375 * 8 =v

v = 27 cubic units

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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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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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Answer:

5cm. 8cm

Step-by-step explanation:

Label all the vertices then use Pythagorean theorem

c² = a² + b²

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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Answer:i don't know I just need the points

Step-by-step explanation:

If a function f(w) = w³ + 2w² - 11w - 12 is divided by (w - 3) then the remainder would be 0.

What is a  polynomial long division?

Polynomial long division is an algorithm in algebra for dividing a polynomial by another polynomial of the same or lower degree, which is a generalized version of the well-known arithmetic technique known as long division. It is simple to do by hand because it divides an otherwise complex division problem into smaller ones.

The given polynomial function is  [tex]f(w)=w^3+2w^2-11w-12[/tex].

And we have to divide the above function by (w - 3).

                                     [tex]=\frac{w^3+2w^2-11w-12}{w-3}\\\\=w^2+\frac{5w^2-11w-12}{w-3}\\\\=w^2+5w+\frac{4w-12}{w-3}\\\\=w^2+5w+4(\frac{w-3}{w-3})\\\\=w^2+5w+4[/tex]

Hence, after taking division, the remainder would be 0.

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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 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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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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