For what value of x is the rational expression below equal to zero?
X-4
(x+5)(x-1)
IOA. 4
OB. 1
O C. -4
OD. -5

The denominator cannot be zero, so

[tex]e^x - 9 = 0 \implies e^x = 9 \implies x = \ln(9)[/tex]

is not in the domain of [tex]f(x)[/tex].

[tex]\ln(x)[/tex] is defined only for [tex]x>0[/tex], and we have

[tex]\ln(x+1) > 0 \implies e^{\ln(x+1)} > e^0 \implies x+1 > 1 \implies x>0[/tex]

so there is no issue here.

By the same token, we need to have

[tex]x+1 > 0 \implies x > -1[/tex]

Taking all the exclusions together, we find the domain of [tex]f(x)[/tex] is the set

[tex]\left\{ x \in \Bbb R \mid x > 0 \text{ and } x \neq \ln(9)\right\}[/tex]

or equivalently, the interval [tex](0,\ln(9))\cup(\ln(9),\infty)[/tex].

The slopes of the two diagonals of the rhombus are: A. b - f/a - e and -a + e/b - f.

To find the slope, the formula used is: change in y/change in x.

If two lines are perpendicular to each other, their slope values will be negative reciprocals.

In a rhombus, the two diagonals in the rhombus bisects each other at angle 90 degrees. This means that the two diagonals of a rhombus are perpendicular to each other. Therefore, the slope of the diagonals of any rhombus would be negative reciprocals.

The slope of the diagonals of the rhombus given would therefore be negative reciprocals to each other.

Given two endpoints of one of the diagonals as:

(a, b) = (x1, y1)

(e, f) = (x2, y2)

Slope (m) = change in y / change in x = b - f/a - e

The negative reciprocal of b - f/a - e is -a + e/b - f.

Therefore, the slopes of the two diagonals are: A. b - f/a - e and -a + e/b - f.

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Answer: -3 ≤ x ≤ -1

Step-by-step explanation:

1 ≤ 3 - 4x ≤ 9

1 + 3 ≤ - 4x ≤ 9 + 3; Add 3 on all sides

4 ≤ -4x ≤ 12

1 ≤ -x ≤ 3; Divide 4 on all sides

-1 ≥ x ≥ -3; Multiply -1 on all sides(FYI: When multiplying or dividing negative numbers in inequalities, make sure to reverse the signs as well)

Answer:

2

Step-by-step explanation:

1≤3-4x≤9

subtract 3

1-3≤3-4x-3≤9-3

-2≤-4x≤6

divide by 2

-1≤-2x≤3

multiply by -1

1≥2x≥-3

or

-3≤2x≤1

divide by 2

[tex]-\frac{3}{2} \leq \frac{2x}{2} \leq \frac{1}{2} \\-\frac{3}{2} \leq x\leq \frac{1}{2} \\[/tex]

length of interval

[tex]=\frac{1}{2} -(\frac{-3}{2} )\\=\frac{1}{2} +\frac{3}{2} \\=\frac{1+3}{2} \\=\frac{4}{2} \\=2[/tex]

Answer: B

Step-by-step explanation:

The horizontal change is 6.

The vertical change is 5.

So, the distance is [tex]\sqrt{6^2 + 5^2}=\sqrt{61}[/tex]

Step-by-step explanation:

let the younger person be x and older be y

4x + 9 = 24

4x = 15

x = 3.75

y = 24 - 3.75

y = 20.25

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◉ [tex]\large\bm{ -4}[/tex]

[tex]\huge\underline{\underline{\boxed{\mathbb {SOLUTION:}}}}[/tex]

Before performing any calculation it's good to recall a few properties of integrals:

[tex]\small\longrightarrow \sf{\int_{a}^b(nf(x) + m)dx = n \int^b _{a}f(x)dx + \int_{a}^bmdx}[/tex]

[tex]\small\sf{\longrightarrow If \: a \angle c \angle b \Longrightarrow \int^{b} _a f(x)dx= \int^c _a f(x)dx+ \int^{b} _c f(x)dx }[/tex]

So we apply the first property in the first expression given by the question:

[tex]\small \sf{\longrightarrow\int ^3_{-2} [2f(x) +2]dx= 2 \int ^3 _{-2} f(x) dx+ \int f^3 _{2} 2dx=18}[/tex]

And we solve the second integral:

[tex]\small\sf{\longrightarrow2 \int ^3_{-2} f(x)dx + 2 \int ^3_{-2} f(x)dx = 2 \int ^3_{-2} f(x)dx + 2 \cdot(3 - ( - 2)) }[/tex]

[tex]\small \sf{\longrightarrow 2 \int ^3_{-2} f(x)dx + 2 \int ^3_{-2} 2dx = 2 \int ^3_{-2} f(x)dx + 2 \cdot5 = 2 \int^3_{-2} f(x)dx10 = }[/tex]

Then we take the last equation and we subtract 10 from both sides:

[tex]\sf{{\longrightarrow 2 \int ^3_{-2} f(x)dx} + 10 - 10 = 18 - 10}[/tex]

[tex]\small \sf{\longrightarrow 2 \int ^3_{-2} f(x)dx = 8}[/tex]

And we divide both sides by 2:

[tex]\small\longrightarrow \sf{\dfrac{2 { \int}^{3} _{2} }{2} = \dfrac{8}{2} }[/tex]

[tex]\small \sf{\longrightarrow 2 \int ^3_{-2} f(x)dx=4}[/tex]

Then we apply the second property to this integral:

[tex]\small \sf{\longrightarrow 2 \int ^3_{-2} f(x)dx + 2 \int ^3_{-2} f(x)dx + 2 \int ^3_{-2} f(x)dx = 4}[/tex]

Then we use the other equality in the question and we get:

[tex]\small\sf{\longrightarrow 2 \int ^3_{-2} f(x)dx = 2 \int ^3_{-2} f(x)dx = 8 + 2 \int ^3_{-2} f(x)dx = 4}[/tex]

[tex]\small\longrightarrow \sf{2 \int ^3_{-2} f(x)dx =4}[/tex]

We substract 8 from both sides:

[tex]\small\longrightarrow \sf{2 \int ^3_{-2} f(x)dx -8=4}[/tex]

• [tex]\small\longrightarrow \sf{2 \int ^3_{-2} f(x)dx =-4}[/tex]

Answer:30%

Step-by-step explanation:

10% of 50 is 50/10 so 10% is 5
15/5=3 so its three lots of 10%

3*10%=30%

Answer:

675

Step-by-step explanation:

100%-32%=68%

459/0.68= 675

Hope this helps! :)

Answer:

See below

Step-by-step explanation:

Distance = rate * time

              = 65 m/hr * 2 1/4 hr = 146.25 miles

rate = distance / time

for Declan :   rate =  146.25 miles / (2 hrs + 35/60 min) = 56.61 mph

The solution for the simultaneous equations 2x + y - 2z = 12, x + 2y +z =18 and 2x - y + 2z =16 are x = 7, y = 4 and z = 3

Linear equations are equations that have constant average rates of change, slope or gradient

A system of linear equations is a collection of at least two linear equations.

In this case, the system of equations is given as

2x + y - 2z = 12,

x + 2y +z =18

2x - y + 2z =16

Eliminate y and z in the equations by adding the first and the third equation together

This gives

2x + y - 2z = 12,

+

2x - y + 2z =16

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

4x = 28

Divide both sides by 4

x = 7

Substitute x = 7 in x + 2y +z =18 and 2x - y + 2z =16

7 + 2y +z =18

2(7) - y + 2z =16

This gives

2y + z = 11

-y + 2z= 2

Multiply -y + 2z= 2 by 2

-2y + 4z= 4

Add -2y + 4z= 4 and 2y + z = 11

5z = 15

Divide by 5

z = 3

Substitute z = 3 in -y + 2z= 2

-y + 2*3= 2

Evaluate

y = 4

Hence, the solution for the simultaneous equations 2x + y - 2z = 12, x + 2y +z =18 and 2x - y + 2z =16 are x = 7, y = 4 and z = 3

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The equations of the three altitudes of triangle ABC include the following:

A triangle can be defined as a two-dimensional geometric shape that comprises three (3) sides, three (3) vertices and three (3) angles only.

A slope is also referred to as gradient and it's typically used to describe both the ratio, direction and steepness of the function of a straight line.

Mathematically, the slope of a straight line can be calculated by using this formula;

[tex]Slope, m = \frac{Change\;in\;y\;axis}{Change\;in\;x\;axis}\\\\Slope, m = \frac{y_2\;-\;y_1}{x_2\;-\;x_1}[/tex]

Also, the point-slope form of a straight line is given by this equation:

y - y₁ = m(x - x₁)

Assuming the following parameters for triangle ABC:

For the equation of altitude AM, we have:

Slope of BC = (2 - 8)/(4 - 0)

Slope of BC = -6/4

Slope of BC = -3/2

Slope of AM = -1/slope of BC

Slope of AM = -1/(-3/2)

Slope of AM = 2/3.

The equation of altitude AM is given by:

y - y₁ = m(x - x₁)

y - 0 = 2/3(x - (-2))

3y = 2(x + 2)

3y = 2x + 4

3y - 2y - 4 = 0.

For the equation of altitude BN, we have:

Slope of CA = (2 - 0)/(4 - (-2))

Slope of CA = 2/6

Slope of CA = 1/3

Slope of BN = -1/slope of CA

Slope of BN = -1/(1/3)

Slope of BN = -3.

The equation of altitude BN is given by:

y - y₁ = m(x - x₁)

y - 8 = -3(x - 0)

y - 8 = -3x

y + 3x - 8 = 0.

For the equation of altitude CL, we have:

Slope of AB = (8 - 0)/(0 - (-2))

Slope of AB = 8/2

Slope of AB = 4

Slope of CL = -1/slope of AB

Slope of CL = -1/4

The equation of altitude CL is given by:

y - y₁ = m(x - x₁)

y - 2 = -1/4(x - 4)

4y - 2= -(x - 4)

4y - 2= -x + 4

4y + x - 2 - 4 = 0.

4y + x - 6 = 0.

In conclusion, we can infer and logically deduce that the equations of the three altitudes of triangle ABC include the following:

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Answer: 40

Step-by-step explanation:

Sub x=4 into 3x²-2x+1

∴ 3(4)²-2(4)+1

=3(16)-8+1

=48-8+1

=40

Answer:

41

Step-by-step explanation:

substitute the x to 4

so 3(4)² - 2(4) + 1 = 41

The dimensions of the most economical shed are 9 ft in length, 9 ft in width, and 9 ft in height if the storage shed is to be built in the shape of a box with a square base.

It is defined as a three-dimensional space enclosed by an object or thing.

It is given that:

A storage shed is to be built in the shape of a box with a square base.

It is to have a volume of 729 cubic feet.

Let x be the length of the base

Let y be the height of the box.

V = 729 cubic feet

The base is square:

l = w = x

V = x²y

y =  729/x²

Cost of the base = $8x²

Cost of the roof = 3x²

Cost of the sides = 4(5.50)xy

= 22xy

Total cost

C= 8x² + 3x² + 22xy

C = 11x² + 22x(729/x²)

C = 11x² + 16038/x

Find the first derivative:

dC/dx = 22x - 16038/x²

Equate; dC/dx = 0

22x - 16038/x² = 0

22x = 16038/x²

22x³ =16038

x³ = 729

x = 9

y = 729/(9)² = 9

Length of base = 9 ft

Width of base  = 9 ft

Height of box = 9 ft

Thus, the dimensions of the most economical shed are 9 ft in length, 9 ft in width, and 9 ft in height.

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Using the law of sines, it is found that the distance between the kite and the dog is of 284.77 meters.

Suppose we have a triangle in which:

The lengths and the sine of the angles are related as follows:

[tex]\frac{\sin{A}}{a} = \frac{\sin{B}}{b} = \frac{\sin{C}}{c}[/tex]

For the situation described, we have that:

Hence:

[tex]\frac{\sin{42^\circ}}{h} = \frac{\sin{87^\circ}}{425}[/tex]

Applying cross multiplication:

[tex]h = 425\frac{\sin{42^\circ}}{\sin{87^\circ}}[/tex]

h = 284.77 meters.

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By solving a linear equation, we conclude that you bought 4 buns and 5 cakes.

We will define the two variables:

We know that you spent exactly £4.35, then we only need to solve the linear equation:

x*0.40 + y*0.55 = 4.35

We need to solve that equation for x and y, such that the values of x and y can only be positive whole numbers.

We can rewrite the equation as:

y*0.55 = 4.35 - x*0.40

y = (4.35 - x*0.40)/0.55

Now we only need to evaluate it in different values of x, and see for which value of x, the outcome y is also a whole number.

We will see that for x = 4, we have:

y = (4.35 - 4*0.40)/0.55 = 5

So you bought 4 buns and 5 cakes.

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40p + 55p = 95p

435p ÷ 95p = 4.57...

So, the sum of the two pairs can go into £4.35 four times.

That meant if I increase the buns & cakes four times, it will help me get closer to the sum of £4.35 & work out which was extra

(40 x 4 = 160) & (55 x 4 = 220)

160 + 220= 380p or £3.80

Adding them up I get 380 pence & that's 55p short to getting to the sum of 435p or £4.35

(435-380=55p)

Thus, the extra is the cake, meaning you bought 4 buns and 5 cakes.

Hope this helps!

Answer: [tex]\pm1,\pm3,\pm5, \pm15[/tex]

Step-by-step explanation:

We can list the possible rational roots of this polynomial using the Rational Root Theorem. This theorem states that all the possible rational roots of an equation follow the structure [tex]\frac{p}{q}[/tex], where p is any of the factors of the constant term and q is any of the factors of the leading coefficient.

In this example, -15 is the constant term and 1 is the leading coefficient ([tex]x^4[/tex] has a coefficient of 1).

The factors of -15 are [tex]\pm1,\pm3,\pm5, \pm15[/tex], while the factors of 1 are [tex]\pm1[/tex]. p is can be any one of the factors of -15, while q can be any of the factors of 1.

[tex]\frac{\pm1,\pm3,\pm5, \pm15}{\pm1}[/tex]

The possible roots can be any of the numbers on the top divided by any of the numbers on the bottom. Since dividing by 1 or -1 won't change any of the numbers on the top, the rational roots of this function are [tex]\pm1,\pm3,\pm5, \pm15[/tex].

Answer:

The volume of the solid is  [tex]1,021m^3[/tex]

Step-by-step explanation:

1. Cylinder

For find the volume of a cylinder we use the next formula:

[tex]V_{cylinder} = \pi h r^2 = \pi \cdot 9m\cdot (5m)^2 \approx 706.86m^3[/tex]

Then the volume of the cylinder is equal to [tex]706.86m^3[/tex]

2. Cones

For find the volume of a cone we use the next formula:

[tex]V_{cone} = \pi \frac{h}{3} r^2 = \pi \cdot \frac{6m}{3} \cdot (5m)^2 \approx 157.08m^3[/tex]

However there are two cones so we have to multiply the volume of one cone for two and we get the total volume for the cones which is [tex]314.16m^3[/tex]

3. Sum of the volumes

Finally we sum the volumes of the cylinder and the cones for get the final result

[tex]V_{cylinder} + V_{cones} = 706.86m^3 + 314.16m^3 \approx 1021m^3[/tex]

So approximating the result is [tex]1021m^3[/tex]

The annual interest rate is 21%

The given parameters are

Loan Amount, P = $300

Interest, I = $63

Number of years, T = 1

The annual interest rate is calculated as

I = PRT

Substitute the known values in the above equation

63 = 300 * R * 1

Evaluate the product

300R = 63

Divide through by 300

R = 21%

Hence, the annual interest rate is 21%

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If you do in fact mean [tex]f(1)=f(6)=0[/tex] (as opposed to one of these being the derivative of [tex]f[/tex] at some point), then integrating twice gives

[tex]f''(x) = -\dfrac1{x^2}[/tex]

[tex]f'(x) = \displaystyle -\int \frac{dx}{x^2} = \frac1x + C_1[/tex]

[tex]f(x) = \displaystyle \int \left(\frac1x + C_1\right) \, dx = \ln|x| + C_1x + C_2[/tex]

From the initial conditions, we find

[tex]f(1) = \ln|1| + C_1 + C_2 = 0 \implies C_1 + C_2 = 0[/tex]

[tex]f(6) = \ln|6| + 6C_1 + C_2 = 0 \implies 6C_1 + C_2 = -\ln(6)[/tex]

Eliminating [tex]C_2[/tex], we get

[tex](C_1 + C_2) - (6C_1 + C_2) = 0 - (-\ln(6))[/tex]

[tex]-5C_1 = \ln(6)[/tex]

[tex]C_1 = -\dfrac{\ln(6)}5 = -\ln\left(\sqrt[5]{6}\right) \implies C_2 = \ln\left(\sqrt[5]{6}\right)[/tex]

Then

[tex]\boxed{f(x) = \ln|x| - \ln\left(\sqrt[5]{6}\right)\,x + \ln\left(\sqrt[5]{6}\right)}[/tex]

The length of the vine is 21 ft

what is the circumference of a circle?

the circumference is the perimeter of a circle

the perimeter of the circle = 2πr = 3 ft

the vine is 7 times the perimeter

length of vine = 7×3= 21 ft

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If the tree is 20 feet tall with circumference 3 feet then the length of  vince is around 21 feet.

Given the height of tree is 20 feet and the circumference of tree is 3 feet.

We have to find the length of vine.

Circumference is the perimeter of a circular object. Because the trunk of a tree is in shape of circle so the perimeter of the trunk is 2πr.

We have been given the circumference of tree be 3 feet.

Circumference=2πr

Because vine is 7 times the circumference so the length of vine being:

Length of vine=7*3

=21 feet.

Hence if the tree is 20 feet tall with circumference 3 feet then the vince is around 21 feet long.

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Answer:

  305 °F

Step-by-step explanation:

The core temperature of the object after 4 hours can be found using an exponential decay formula to model the decay of the difference between core temperature and ambient.

The solution to the differential equation described by Newton's law of cooling is the exponential equation ...

  y = ab^t +c

where 'a' is the initial core temperature difference from ambient, 'b' is the decay factor of that difference in 1 unit of time period t. 'c' is the ambient temperature.

For this problem, the ambient temperature is c=80, and the differences of interest are ...

  a = 1200 -80 = 1120

  b = (830 -80)/1120 = 75/112

Using these values in the model gives ...

  y = 1120(75/112)^t +80 . . . . . . where y(t) is the core temperature at time t

Note that units of time are hours.

We want y when t=4.

  y = 1120(75/112)^4 +80 ≈ 1120(0.20108) +80 ≈ 305.212

The core temperature after 4 hours is about 305 °F.

__

Additional comment

The differential equation will have a solution of the form ...

  [tex]T-T_R=(T_0-T_R)e^{kt}[/tex]

where k = ln(75/112) ≈ -0.40101

In the above, we defined b = e^k = 75/112. Accuracy with this fraction can be better than using a truncated value of k.

The radius of the circle of the circle equation (x + 3/8)^2 + y^2 = 1 is 1 unit

The circle equation of the graph is given as:

(x + 3/8)^2 + y^2 = 1

The general equation of a circle is represented using the following formula

(x - a)^2 + (y - b)^2 = r^2

Where the center of the circle is represented by the vertex (a, b) and the radius of the circle is represented by r

By comparing the equations (x - a)^2 + (y - b)^2 = r^2 and (x + 3/8)^2 + y^2 = 1, we have the following comparison

(x - a)^2 = (x + 3/8)^2

(y - b)^2 = y^2

1 = r^2

Rewrite the last equation as follows:

r^2= 1

Take the square root of both sides of the equation

√r^2 = √1

Evaluate the square root of 1

√r^2 = 1

Evaluate the square root of r^2

r = 1

Hence, the radius of the circle of the circle equation (x + 3/8)^2 + y^2 = 1 is 1 unit

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Answer: Look in step-by-step explanation

Step-by-step explanation:

Area has a unit of cm^2 or m^2, whilst Volume has a unit of cm^3 or m^3

Using this information, we can see that we have to square both sides of the similarity ratio for the area ratio and cube both sides of the similarity ratio for the volume ratio
For example, question 2 states that the similarity ratio is 3:6, so the area ratio is 3^2:6^2 or 9:36 and the volume ratio is 3^3:6^3 = 27:216

You can do the rest from here

Explanation:

There are 25 choices for president. Then we have 25-1 = 24 choices for the secretary, and 24-1 = 23 choices for treasurer. This countdown (25,24,23) is because any given person cannot serve in more than one position.

Multiply out those values: 25*24*23 = 13800

Side note: you could use the nPr permutation formula as an alternative path. Plug in n = 25 and r = 3.

The amount of kilograms of each type of original fruit juice weighs; 12 kg of Apple and 6 kg of grape

The concentrations of apple and grape juice in the two different juice types are given as;

Juice A:

(2/3) apple

(1/3) grape

Juice B:

(1/3) apple

(2/3) grape

Let a be how much juice A is in the mixture;

Let b be how much juice B is in the mix.

Thus;

²/₃a + ¹/₃b = 10     ------(eq 1)

¹/₃a + ²/₃b = 8       ------(eq 2)

Solving both equations simultaneously gives us;

a = 12 and b = 6

Thus, the amount of kilograms of each type of original fruit juice weighs; 12 kg of Apple and 6 kg of grape

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Answer:

If x^2 = 1, then x = 1 is NOT true because x can also be -1.

Answer:

3

Step-by-step explanation:

2+ (-2/3)^2 ÷ 1/3

make -2/3 postive because its being raised by an even exponent

you never divide by fractors so make 1/3 into 3 and make it multiplication

2+ (2/3)^2 × 3

raise the fraction to the power of 2

2+ (4/9)× 3

cancel out the GCF

2+ 4/3

add

10/3

simplify

3.3 ≅ 3

Answer:

Step-by-step explanation:

x² + 2x = x(x + 2)

[tex]\sf \dfrac{x +5}{x + 2}-\dfrac{x+1}{x^2+2x}=\dfrac{x + 5}{x +2}-\dfrac{x+1}{x(x+2)}[/tex]

LCM = x(x+2)

                        [tex]\sf =\dfrac{(x+5)*x}{(x+2)*x}-\dfrac{x+1}{x(x+2)}\\\\=\dfrac{x*x + 5*x}{x^2+2x}-\dfrac{x+1}{x^2+2x}\\\\=\dfrac{x^2+5x - (x+1)}{x^2+2}\\\\=\dfrac{x^2+5x -x - 1}{x^2+2x)}\\\\=\dfrac{x^2+4x-1}{x^2+2x}[/tex]

The decimal notation for the given value in scientific notation is 215400.

Very large or very small numbers are denoted in scientific notation to simplify complex calculations.

The scientific notation is represented by: a × 10ⁿ

Wher a - any number or decimal number

n - the power of 10; it is an integer number

To convert this into decimal notation,

The given number is in the scientific notation,

2.154 × 10⁵

Since the power of 10 is positive, the decimal point in the number 2.154 moves towards the right. I.e,

2.154 × 10⁵ = 215400

Here power is 5. So, the decimal point moved five places to the right.

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Answer: 25 in³

Step-by-step explanation:

We can calculate the volume of the pyramid by first calculating the volume of a prism with the same dimensions, and then dividing by 3 (all pyramids a volume that's [tex]\frac{1}{3}[/tex] of the volume of a prism with the same dimensions).

The volume of a prism is its base area times its height. The base would be a square, so its area is 5², which is 25. The height is 3 inches, making the prism's volume 75 in³.

The volume of the pyramid would be one-third of this value, which is 75[tex]75\div3[/tex] which is 25 in³.