Determine whether the system is consistent or inconsistent and whether the equations are dependent or independent.
-3x + 3y= 21
4x + 4y= 60

Answer:

k)  x = 2

l)  x = 6

Step-by-step explanation:

[tex]\boxed{\begin{minipage}{8cm}\underline{Log laws}\\\\Product law:\quad\:$\log_axy=\log_ax + \log_ay$\\\\Power law:\quad\;\;\;\:$\log_ax^n=n\log_ax$\\\\Equality Law:\; If $\log_ax=\log_ay$ then $x=y$\\\\Same base as number:\;\;\;$\log_aa=1$\\$\end{minipage}}[/tex]

[tex]\begin{aligned}\log_7(x + 12) + \log_7(x + 1) &= \log_7(42)\\\log_7\left[(x + 12)(x+1)\right]&= \log_7(42)\\(x + 12)(x+1)&=42\\x^2+13x+12&=42\\x^2+13x-30&=0\\x^2+15x-2x-30&=0\\x(x+15)-2(x+15)&=0\\(x-2)(x+15)&=0\\\\\implies x&=2\\\implies x&=-15\end{aligned}[/tex]

As logs of negative numbers cannot be taken, x = -15 is an extraneous solution.  Therefore, the only valid solution is x = 2.

[tex]\begin{aligned}\log_4(x + 3) + 1 &= 2\log_4(x)\\\log_4(x + 3) + \log_44 &= \log_4(x^2)\\\log_4\left[4(x + 3)\right] &= \log_4(x^2)\\4(x + 3)&=x^2\\4x+12&=x^2\\x^2-4x-12&=0\\x^2-6x+2x-12&=0\\x(x-6)+2(x-6)&=0\\(x+2)(x-6)&=0\\\\\implies x&=-2\\\implies x&=6\end{aligned}[/tex]

As logs of negative numbers cannot be taken, x = -2 is an extraneous solution.  Therefore, the only valid solution is x = 6.

The reflection preserves the side lengths and angles of the triangle. The dilation preserves angles but not side lengths. Therefore, option B is the correct answer.

Dilation is the process of resizing or transforming an object. It is a transformation that makes the objects smaller or larger with the help of the given scale factor. The new figure obtained after dilation is called the image and the original image is called the pre-image.

Given that, triangle is reflected across the y-axis and then dilated by a factor of centered at the origin.

The triangle ABC in a graph with vertices A (-4, -2), B (4, -2), and C (4, 4)

A reflection across the y-axis changes the coordinates of the triangle from (x, y) to (-x, y) without changing the side lengths and angles.

The dilation of the triangle by a factor an then centered at the origin increases the side length of the triangle but the angles in the triangle remain the same since the ratio of the sides before and after dilation are the same.

Therefore, option B is the correct answer.

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Answer:

a)  Growth rate of 2.737% (3 d.p.) per year

b)  117,897

Step-by-step explanation:

[tex]\boxed{\begin{minipage}{9 cm}\underline{General form of an Exponential Function with base $e$}\\\\$f(t)=ae^{rt}$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the initial value ($y$-intercept). \\ \phantom{ww}$\bullet$ $e$ is Euler's number. \\ \phantom{ww}$\bullet$ $r$ is the continuous growth/decay rate.\\ \phantom{ww}$\bullet$ $t$ is time.\\\end{minipage}}[/tex]

If a > 0 and r > 0, the function is an exponential growth function.

If a > 0 and r < 0, the function is an exponential decay function.

Given exponential function:

[tex]P(t) = 90000\cdot e^{0.027t}[/tex]

As r > 0 the function is an exponential growth function.

The annual growth rate is:

[tex]\begin{aligned}\textsf{Annual growth rate}& = e^{0.027}-1\\&=1.02736780...-1\\&=0.0273678027...\\&=2.73678027...\%\\&=2.737\%\; \sf (3\;d.p.)\end{aligned}[/tex]

To calculate how many residents the model predicts will live in Asheville, NC in 2030, substitute t = 10 into the given function:

[tex]\begin{aligned}\implies P(10) &= 90000\cdot e^{0.027 \cdot 10}\\&= 90000\cdot e^{0.27}\\ &= 90000\cdot 1.30996445...\\&=117896.800...\\&=117897\; \sf (nearest\;whole\;number)\\\end{aligned}[/tex]

Therefore, the model predicts that there will be a population of approximately 117,897 living in Asheville, NC in 2030.

0.1920 mL of the anesthetic should be given to the patient.

First, we need to convert the patient's weight from kilograms to pounds:

58 kg x 2.20462 lb/kg = 128.01 lbs

Next, we can calculate the total amount of anesthetic to be given to the patient:

15 mg/10 lbs x 128.01 lbs = 19.20 mg

Finally, we can divide the total amount of anesthetic by the concentration of the anesthetic in each mL:

19.20 mg / 100 mg/mL = 0.1920 mL

So, 0.1920 mL of the anesthetic should be given to the patient.

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A. An expression for the perimeter of the fountain is 8x + 12 ft.

B. The perimeter when x=2 ft. is 28 ft

The area of the fountain is given as 4x^2 + 12x + 9 square feet. Since the fountain is a square, we know that the length and the width of the fountain are equal. This means that we can express the area of the fountain in the form of (cx + d)^2.

Expanding (cx + d)^2 gives:

(cx + d)^2 = cx^2 + 2cxd + d^2

We can equate this to the given area of the fountain:

cx^2 + 2cxd + d^2 = 4x^2 + 12x + 9

To find the expression for the perimeter of the fountain, we need to find the dimensions (length and width) of the fountain. This can be done by factoring the area expression:

4x^2 + 12x + 9 = (2x + 3)(2x + 3)

So the dimensions of the fountain are (2x + 3) ft

The perimeter of a square is the sum of all four sides, so we can find the perimeter of the fountain by multiplying the length of one side by 4:

Perimeter = 4(2x + 3) ft = 8x + 12 ft

When x = 2 ft, the perimeter of the fountain is:

8(2) + 12 = 28 ft

So when x = 2 ft, the perimeter of the fountain is 28 ft.

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Answer:

[tex]\textsf{a)} \quad K(t)=1500(1.12)^t[/tex]

b)  44940

c)  57.4 days

Step-by-step explanation:

Exponential Growth Function

[tex]f(t)=a(1+r)^t[/tex]

where:

Given values:

To create a function K(t) that expresses the number of users t days after the start of the year, substitute the given values into the exponential growth formula:

[tex]\implies K(t)=1500(1+0.12)^t[/tex]

[tex]\implies K(t)=1500(1.12)^t[/tex]

To calculate how many users there are after 30 days, substitute t = 30 into the function K(t) from part a:

[tex]\implies K(30)=1500(1.12)^{30}[/tex]

[tex]\implies K(30)=44939.8831...[/tex]

[tex]\implies K(30)=44940\; \rm users[/tex]

To calculate after how many days it takes to reach one million users, substitute K(t) = 1,000,000 into the function and solve for t:

[tex]\implies 1000000=1500(1.12)^t[/tex]

[tex]\implies \dfrac{2000}{3}=1.12^t[/tex]

[tex]\implies \ln \left(\dfrac{2000}{3}\right)=\ln 1.12^t[/tex]

[tex]\implies \ln \left(\dfrac{2000}{3}\right)=t \ln 1.12[/tex]

[tex]\implies t=\dfrac{\ln \left(\frac{2000}{3}\right)}{\ln 1.12}[/tex]

[tex]\implies t=57.3755016...[/tex]

[tex]\implies t=57.4\; \rm days[/tex]

Answer:

19.8

Step-by-step explanation:

Answer:

Do 99 divided by 5

Step-by-step explanation:

Use cross multiplication

The table is completed as follows:

The situation is modeled by a linear function, in slope-intercept format, as follows:

y = mx + b.

In which:

The initial capacity is of 770 liters, decaying by 14 liters each minute, hence the parameters are given as follows:

b = 770, m = -14.

The function for this problem is defined as follows:

y = -14x + 770.

The numeric values are given as follows:

The table is given by the image presented at the end of the answer.

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The simplified expression is 14m -10n

From the question, we have the following parameters that can be used in our computation:

12m−3n62m−7n

Express properly

So, we have the following representation

12m - 3n + 2m - 7n

Collect the like terms

This gives

12m + 2m - 3n - 7n

Evaluate the like terms

So, we have the following representation

14m - 10n

Hence, the solution is 14m - 10n

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The linear function containing the median of the trapezoid whose vertices are R(-1, 5) , S(1, 8), T(7, -2), and U(2, 0). is given as follows:

y = -1.67x + 6.5.

The slope-intercept definition of a linear function is given as follows:

y = mx + b.

In which:

The median of the trapezoid is composed by these two points:

The coordinates of the midpoint of RS are given as follows:

The coordinates of the midpoint of TU are given as follows:

Hence the points of the function are:

(0, 6.5) and (4.5, -1).

When x = 0, y = 6.5, hence the intercept b is given as follows:

b = 6.5.

When x increases by 4.5, y decays by 7.5, hence the slope m is given as follows:

m = -7.5/4.5

m = -1.67.

Hence the function is:

y = -1.67x + 6.5.

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The mean of the sample is the random variable in the given problem.

The mean of any given data set can be defined as summarizing a given group of data. It is done in order to make data sets and their values easier to understand. It can be obtained by first adding all the given values and then dividing the total sum by the number of data entries.

A random variable is any quantity that depends on random events. Here, we know that the mean of the given sample depends on all of the random events or we can say random values present in the table. Hence, we can conclude that the random variable is the mean of the sample.

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The required constant charge for each minute used is $0.02.

Algebraically speaking, an equation is a statement that shows the equality of two mathematical expressions. For instance, the two equations 3x + 5 and 14, which are separated by the 'equal' sign, make up the equation 3x + 5 = 14. ​

Let fixed charge of monthly be x and constant rate of each minute used is y.

So,

In january

x + 100y = 47--------(1)

and in February

x + 350y = 52-----------(2)

equation(2) - equation(1)

(x + 350y) - (x + 100y) = 52 -47

250y = 5

y = 0.02

Thus, $0.02 used per minutes.

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If y varies inversely as x and y = 9 when x = 7, then the value of y = 10.5 when x = 6.

If y fluctuates inversely of x, then y = kx, where k is a fixed value. According to the expressions "y varies inversely as x" and "y is inversely proportionate to x," y decreases as x increases or vice versa. There are two translations for this idea. For some constant k, referred to as the constant of proportionality, yx = k.

According to the expression "y varies inversely as x," y drops by the same amount that x grows. In other terms, xy = k means that the equation xy is constant.

Let the equation be x = k/y, then k be a constant

[tex]$$\begin{aligned}& x=k \times \frac{1}{y} \\& \Rightarrow k=x y=9 \times 7=63 \\& \Rightarrow x=\frac{63}{y}\end{aligned}$$[/tex]

So [tex]$y=\frac{63}{x}$[/tex]

At x = 6 we have

[tex]$y=\frac{63}{6}=10.5$$[/tex]

If y varies inversely as x and y = 9 when x = 7, then the value of y = 10.5 when x = 6.

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The maximum possible number of points where two polygons can intersect is [tex]$n = 4$[/tex]. This is achieved when the two polygons are tangent to each other at four points.

There are different ways to approach the problem of determining the maximum possible number of points where two polygons can intersect, but one possible method is to use geometric principles and constructive examples. Here is one possible way to do it:

This is a constructive example for showing that the maximum possible number of intersection points between heptagon and hexagon is 4.

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Answer: b is the coefficient of the x term. In the standard form f(x) = ax^2 + bx + c, b is the value of the coefficient of the x term. So the value of b is 0.

Step-by-step explanation:

The value of b in the given function is 12.

An equation is a mathematical statement that shows that two mathematical expressions are equal.

For example, 3x + 5 = 14 is an equation, in which 3x + 5 and 14 are two expressions separated by an 'equal' sign.

Given that, a table defining the function, we need to form an equation and compare with the standard form of a quadratic function,

We can use the vertex form of a quadratic function, which is f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola.

Since, we know the vertex of this parabola from the table, we can use that information to write the equation in vertex form: f(x) = a(x + 6)^2 + 8

Expanding this, we get f(x) = a(x² + 12x + 36) + 8

Comparing this with the standard form, we see that a = 1, b = 12, and c = 8.

Therefore, the value of b is 12.

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The complete question is :-

The table represents a quadratic function f(x). x f(x) −10 24 −9 17 −8 12 −7 9 −6 8 −5 9 −4 12 If the equation of the function f(x) is written in standard form f(x) = ax2 + bx + c, what is the value of b?

Answer:

x = 15

Step-by-step explanation:

The sum of the four angles of a quadrilateral add up top 360°

Given:

m∠A = (5x+14)°  

m∠b = 109°        

m∠C = (3x+8)°    

m∠D = 109°°

Here we are given actual values for two angles as 109°. Their sum is: 109 + 109 = 218°

The sum of the remaining two angles which are expressions in x should be:

360 - 218 = 142°

This is the sum of m∠A an m∠C

Substituting the terms for the two angle measures we get

(5x+14)  + (3x+8) = 142

5x+14  + 3x+8 = 142

Grouping like terms:

5x + 3x + 14 + 8 = 142

8x + 22 = 142

8x = 142- 22

8x = 120

x = 120/4

x = 15

890998

Step-by-step explanation:

1999+888999 = 890998

The length of sides a,b and c of triangle are 2, 4 and 5 respectively whereas the perimeter is 11ft.

A one-dimensional length, a two-dimensional shape, or both can have perimeters that are closed paths that contain, surround, or define them. A circle's or an ellipse's perimeter is measured by its circumference.

The perimeter calculation has a lot of useful applications. Calculated perimeter is the term used to describe the size of fence required to completely surround a yard or garden. How far a wheel or circle can go in one spin is determined by its circumference, which measures its perimeter. Similar to this, a spool's circumference is inversely related to the length of thread that is wound around it; if the string were exactly the same length, the circumference would be equal.

Given that

b = 2a

c = 3 +a

We know that the perimeter is equal to the sum of all the sides

Perimeter = a+b +c

11 = a +2a + a+3

8 = 4a

a = 2

Hence, sides a ,b and c are 2, 4, 5 respectively.

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Answer:

(1.1)

1000g=1kg

500÷1000=0.5kg

(1.2)

1000m=1km

1000×1000= 1000000m

(1.3)

1000kg=1tonne

165×1000= 165,000kg

Answer:

All of the following statements are true for the expression (11 + 7)(8 + 7) + (2 + 9): all factors are the sum of two terms, the solution is the sum of three terms, the solution is the product of two factors, plus another number, and the solution is the difference between three terms.

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The length of the arc of a circle with given radius and central angle is 5.23 meters.

A circle is a two-dimensional figure formed by a set of points that are at a constant or at a fixed distance (radius) from a fixed point (center) on the plane. The fixed point is called the origin or center of the circle and the fixed distance of the points from the origin is called the radius.

Given that, a circle of diameter 10 meters subtended by a central angle of π/3 radians or 60°.

We have, radius of a circle = 5 meters

We know that, arc length of a circle is θ/360° ×2πr

Now, 60°/360° ×2×3.14×5

= 1/6×2×3.14×5

= 5.23 meters

Therefore, the length of the arc of a circle with given radius and central angle is 5.23 meters.

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The oscillations have an amplitude of 0.129 m. The block is positioned at 0m and moving at 0m/s at time t=0s.

The maximum deflection of a simple harmonic oscillator from its equilibrium position is its amplitude. Since this is the displacement at time t=1.00s, the oscillations in this instance have an amplitude of 0.129m. The oscillator is in equilibrium at time t=0 s, therefore the block is located at location 0 m. Additionally, since the oscillator has not yet begun to move, these values can be determined by examining the oscillator's equation of motion. According to this equation, the oscillator's position is determined by the expression x=A*cos(t), where A stands for amplitude and for angular frequency. The equation becomes simpler to x=0m for A=0.129m and t=0s. Similar to that, v=-Asin(t) gives the velocity. The equation is again simplified to v=0m/s as A=0.129m and t=0s.

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The missing coordinates on the ordered pairs of the linear equation are:

(0, -4) and  (8, 3)

Here we have the linear equation:

3x - 4y = 12

And we want to find the missing coordinates for the given points (0, ?) and (?, 3) where these points belong to the line.

The first point is (0, ?), this means that the value of x is zero, then we can evaluate the equation in x = 0 then we will get:

3*0 - 4y = 12

-4y = 12

y = 12/-3 = -4

So the point is (0, -4)

For the other point (?, 3) we need to evaluate in y = 3.

3*x - 4*3  12

3x - 12 = 12

3x = 12 + 12

3x = 24

x = 24/3 = 8

So we have the point (8, 3)

These are the missing coordinates.

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A. The percent error is 25%

B. Another estimate that gives the same percent error is the Theoretical value.

A. To find the percent error, we use the following formula:

percent error = (|experimental value - theoretical value| / theoretical value) x 100

In this case, the experimental value is 4000 years (from carbon-14 dating) and the theoretical value is 3200 years (from the researcher's estimate). Plugging these values into the formula:

percent error = (|4000 - 3200| / 3200) x 100 = (800 / 3200) x 100 = 25%

B. To find an estimate that gives the same percent error, we can use the formula:

Theoretical value = experimental value / (1 + percent error / 100)

If we use the same percent error (25%) and experimental value (4000 years) as before:

Theoretical value = 4000 / (1 + 25/100) = 4000 / 1.25 = 3200

So, an estimate of 3200 years would give the same 25% percent error as the researcher's original estimate.

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The measure of a different interior angle of the triangle is 52 degrees.

A right angled triangle is a triangle with one of the angles as 90 degrees.

The sum of the interior angle of a triangle is 180 degrees.

Given the data in the question;

Since the sum of the interior angle of a triangle is 180 degrees.

Angle a + Angle b + Angle c = 180

38 + Angle b + 90 = 180

Angle b + 128 = 180

Angle b = 180 - 128

Angle b = 52 degrees.

Therefore, the measure of the second angle is 52 degrees.

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The amount you make each month before taxes and other deductions is known as your monthly gross income so monthly gross income is 3000.

Marcus makes $3,000 each month. He pays a total of 28% in taxes and 5% for

It's rather easy to figure out net income. Simply deduct expenses like taxes, insurance, and retirement contributions from your gross income, which is the entire amount of money you have made. Deductions from gross income equal net income.

Gross income – deductions = net income.

3000 - 840 - 150 = 2010.

The amount you make each month before taxes and other deductions is known as your monthly gross income. In other words, it is your annual income divided by 12.

monthly gross income = 3000.

The amount you make each month before taxes and other deductions is known as your monthly gross income so monthly gross income is 3000.

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The distance of IH is 1.089 meters.

Circle is a two dimensional figure which consist of set of all the points which are at equal distance from a point which is fixed called the center of the circle.

Given that,

Radius = 5.1 meters

EF = EI = 5.1

FG = 6.3 meters

HE is a perpendicular bisector of FG.

FH = HG = 6.3 / 2 = 3.15

Also, FH is the projection of EF in the vertical axis.

FH = EF sin (θ), where θ is the angle FEH.

3.15 = 5.1 sin (θ)

sin (θ) = 0.6176

θ = sin⁻¹ (0.6176)

θ = 38.14°

IH = EI - EH

We have,  EH is the projection of EF into the horizontal axis.

EH = EF cos (θ)

Substituting,

IH = 5.1 -  EF cos (θ)

IH = 5.1 - 5.1 cos (38.14)

IH = 5.1 ( 1 - cos (38.14))

IH = 1.089 meters

Hence the length of the line segment IH is 1.089 meters.

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Using Angle Sum property, the measure of 3 angles is ∠Q = 35°, ∠R = 19°, ∠S = 126°.

According to the triangle's "angle sum property," a triangle's angles add up to 180 degrees. Three sides and three angles, one at each vertex, make up a triangle. The sum of the interior angles in a triangle is always 180o, regardless of whether it is acute, obtuse, or right.

One of the most commonly applied properties in geometry is the triangle's angle sum property. Most often, the unknown angles are calculated using this attribute.

As given the measure of 3 angles is -

∠Q = 4x - 17

∠R = x + 6

∠S = 10x - 4

We know that by angle - sum property,

The sum of 3 angles of triangle = 180°

⇒ 4x - 17 + x + 6 + 10x - 4 = 180

⇒ 15x -15 = 180

⇒ 15x = 180 + 15

⇒ 15x = 195

⇒ x = 13

∠Q = 4x - 17 = 4 × 13 - 17 = 35°

∠R = x + 6 = 13 + 6 = 19°

∠S = 10x - 4 = 10 × 13 - 4 = 126°

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Complete question -