Solve the following exponential equation. Express the solution set in terms of natural logarithms or common logarithms.​ Then, use a calculator to obtain a decimal approximation for the solution.

e^2x-7 -5 = 27132
The solution set expressed in terms of logarithms is {___}
The Solution set is {___}

The rate of change between 1992 and 2006 is a decrease of 8.39%.

To find the annual rate of change between 1992 and 2006, we can use the formula:

r = [tex](V2/V1)^{1/n}[/tex] - 1

where V1 is the initial value, V2 is the final value, and n is the number of years between the two values.

r = -0.0839

Therefore, the annual rate of change between 1992 and 2006 is -0.0839.

To express the rate of change in percentage form, we can multiply the result from part A by 100:

r = -0.0839 x 100

r = -8.39%

Therefore, the rate of change between 1992 and 2006 is a decrease of 8.39%.

To find the value of the car in the year 2009, we can assume that the value continues to drop at the same percentage rate as calculated in part A.

From 2006 to 2009, there are 3 years. So, using the formula for exponential decay, we have:

where V0 is the value in 2006, r is the rate of decrease, and n is the number of years between 2006 and 2009.

V = 11792.51

Therefore, the value of the car in the year 2009 would be approximately $11,800 (rounded to the nearest $50).

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There are 8 possible outcomes and we need to find the probability of having at least two male children.

Out of the 8 outcomes, there are 3 outcomes where all children are female (FFF, probability = 1/8) and 3 outcomes where there is exactly one male child (FMF, FFM, MFF, probability = 3/8).

Therefore, the probability of having at least two male children is the complement of these outcomes, which is 1 - (1/8 + 3/8) = 4/8 or 1/2.

So the probability of selecting a family with at least two male children is 1/2 or 0.5.

The probability of selecting a family with at least two male children is 3/8.

It is the chance of an event to occur from a total number of outcomes.

The formula for probability is given as:

Probability = Number of required events / Total number of outcomes.

Example:

The probability of getting a head in tossing a coin.

P(H) = 1/2

We have,

There are 8 equally likely outcomes, and we want to find the probability of selecting a family with at least two male children.

There are three outcomes that satisfy this condition:

MMM, MMF, and MFM.

The probability of each of these outcomes is:

MMM: (1/2) x (1/2) x (1/2) = 1/8

MMF: (1/2) x (1/2) x (1/2) = 1/8

MFM: (1/2) x (1/2) x (1/2) = 1/8

So the total probability of selecting a family with at least two male children is:

1/8 + 1/8 + 1/8 = 3/8

Therefore,

The probability of selecting a family with at least two male children is 3/8.

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The 5-digit number that can be formed is 100,000.

5-digit numbers using 0,1,,2,3,4,5,6,7,8,9,

 We can use any of the 10 numbers to form a 5-digit number

The first place, there can be any of the 10 numbers

       In the second place, there can be any of the 10 numbers

       In the third place, also there can be any of the 10 numbers and so on.

       Therefore we have:

   10×10×10×10×10 = 100,000 digits

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Using the method of conversion,

The patient will take 6 teaspoons of the elixir and 2 tablespoons of the elixir as prescribed by the physician.

Conversion means changing a particular unit of measurement into another so that the values or the measurements are in a single unit which then allows us to solve the problem or the sum easily. Both the base units compared should be same.

This helps to reduce the complications that are created in a question.

Now here in the question,

We have 1 ounce of elixir.

Now to convert 1 ounce in 1 teaspoon we need to multiply the value by 6.

Similarly in order to convert 1 ounce into tablespoon, we need to multiply it by 2.

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Answer: The dimensions on the map are 10 cm by 15 cm, so the area on the map is 10 cm × 15 cm = 150 cm^2.

Since the scale is 1 cm = 50 m, every 1 cm on the map corresponds to an actual distance of 50 m. Therefore, the actual dimensions of the park are 10 cm × 50 m/cm = 500 m by 15 cm × 50 m/cm = 750 m.

The actual area of the park is the product of the actual dimensions, so the actual area is 500 m × 750 m = 375000 square meters.

Step-by-step explanation:

Answer:

1/3

Step-by-step explanation:

We can first find 7/9 of 3/7 by multiplying 7/9 and 3/7:

[tex]\frac{7}{9}*\frac{3}{7}=\frac{21}{63}[/tex]

Now, we can simplify the fraction by finding the GCF of the numerator and 63.

The GCF shared by 21 and 63 is 21, as 21 is the highest number that both the numerator (21) and the denominator (63) are evenly divisible by.  Thus, we divide both the numerator and denominator by 21 to simplify the fraction fully and bring it to lowest terms:

[tex]\frac{21/21}{63/21}=\frac{1}{3}[/tex]

Answer:

[tex]x = ± \frac{5 \sqrt{3} }{3} [/tex]

Step-by-step explanation:

[tex](x + 3)(x - 3) = 16 - 2 {x}^{2} [/tex]

Use the quick multiplication formula:

[tex] {x}^{2} - 9 = 16 - 2 {x}^{2} [/tex]

[tex] {x}^{2} + 2 {x}^{2} = 16 + 9[/tex]

[tex]3 {x}^{2} = 25[/tex]

Divide both parts by 3 to make x the subject:

[tex] {x}^{2} = ±8 \frac{1}{3} [/tex]

[tex]x = ± \frac{5 \sqrt{3} }{3} [/tex]

Scenario A: Cannot be determined

Scenario B: Linear

Scenario C: Cannot be determined

Given triangle is not a right angle triangle.

What is Pythagoras theorem?

For any right angle triangle,

Hypotenuse ²= Base²+Height ²

To determine whether the any triangle is a right triangle, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Let's denote the sides of the triangle as follows,

a = 14

b = 10

c = 4√19

Then we can check whether the Pythagorean theorem holds for these values:

c² = (4√19)² = 16×19 = 304

a²+ b² = 14²+ 10² = 196 + 100 = 296

Since c² is not equal to a² + b², the triangle is not a right triangle. Therefore, the answer is no, the given triangle is not a right angle triangle.

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Step-by-step explanation:

Numbers:

[tex]1 \frac{5}{8} [/tex]

[tex] - 2.35[/tex]

[tex] - 4.3[/tex]

[tex] - \frac{3}{5} = - 0.6[/tex]

.

From the least to the greatest:

[tex] - 4.3[/tex]

[tex] - 2.35[/tex]

[tex] - 0.6[/tex]

[tex]1 \frac{5}{8} [/tex]

The calculated values of the angles 1 and 2 are <1 = 95.9 and <2 = 56.8

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

The triangle

The adjacent angle of 117.5 is

Angle = 180 - 117.5

Angle = 62.5

The third angle in the big triangle is

Angle = 180 - 62.5 - 39.1

Angle = 78.4

So, the measure of angle 2 is calculated as

<2 = 180 - 21.6 - (180 - 78.4)

<2 = 56.8

Also, the measure of angle 1 is calculated as

<1 = 39.1 + <2

<1 = 39.1 + 56.8

<1 = 95.9

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Answer:

90

Step-by-step explanation:

We can translate this into the following equation:

x - 0.4x = 54

Simplifying the left side of the equation, we get:

0.6x = 54

Dividing both sides by 0.6, we get the following:

x = 90

Therefore, 90 is the number decreased by 40% leaving 54.

The proof for the above question are given above accordingly.

10) Since OA ⊥ OC; and

OB ⊥ OD (Given)
Thus,

∠AOC = 90°;  perpendicular bisector theorem

∠BOD = 90°; perpendicular bisector theorem

Therefore,

90° Less ∠1 = ∠BOC; and

90° less ∠3 = ∠BOC,

Thus,

∠1≅∠3 - subtraction theorem of congruence

11)

Since ∠A is complimentary to ∠ ADB; Given.....1

and ∠C is complimentary to ∠CDB - Given. .....2
Where DB  bisects ∠ADC, thus,

∠DBC = 90° ----Line bisector theorem .......3
∠BDC = 90° -----Line bisector theorem........4

From 1 and 2 above, we know that:

∠A + ∠ADB = 90°
∠C + ∠CDB = 90°, thus,


Since

∠A + ∠ADB + ∠DBA = 180° - Sum of Angles in a triangle; and

∠C + CDB + ∠BDC = 180° - Sum of angles in a triangle, we can state that

∠A ≅∠C.

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Jackson will have earned $72 in interest at the end of 3 years.

Jackson saves $25 per month for 12 months:

$25 * 12 months = $300 per year

Now, let's calculate the total amount saved at the end of each year:

Year 1: $300

Year 2: $300 + $300 = $600

Year 3: $300 + $600 = $900

Next, we'll apply the 4% interest rate compounded annually to the final yearly balance:

Year 1: $300 * (1 + 0.04) = $300 * 1.04 = $312

Year 2: $600 * (1 + 0.04) = $600 * 1.04 = $624

Year 3: $900 * (1 + 0.04) = $900 * 1.04 = $936

So, the total amount Jackson will save at the end of 3 years is:

$312 (Year 1) + $624 (Year 2) + $936 (Year 3) = $1,872

Total deposited:

$300 + $600 + $900

= $1,800

Total interest earned:

$1,872  - $1,800

= $72

So, Jackson will have earned $72 in interest at the end of 3 years.

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According to the solution we have come to find that, As the metal lid was heated, it expanded so that it was not as tight. The correct answer is (G) As the metal lid was heated, it expanded so that it was not as tight.

what is heat?

Heat is a form of energy that is transferred between objects or systems as a result of a temperature difference. It is the energy that flows from a body of higher temperature to a body of lower temperature until they reach a state of thermal equilibrium. Heat can be transferred through three different mechanisms: conduction, convection, and radiation.

Conduction involves the transfer of heat through direct contact between two objects that are at different temperatures. Convection involves the transfer of heat through the movement of fluids, such as air or water, due to differences in temperature. Radiation involves the transfer of heat through the emission and absorption of electromagnetic waves, such as infrared radiation.

The correct answer is (G) As the metal lid was heated, it expanded so that it was not as tight.

When the metal lid was placed in hot water, the heat caused it to expand. This expansion allowed the lid to loosen from the glass jar, making it easier to turn. This is because metals generally expand when heated and contract when cooled. The expansion of the metal lid due to heat causes it to become slightly larger than before, thereby loosening its grip on the jar. As a result, the lid can be turned with less force, and it comes off easily.

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Answer:

The total volume, in cubic feet, of the three boxes is 144 cubic feet.

Chain of thought:

1. Volume = base x height

2. Box 1: Volume = 16 sq feet x 3 feet = 48 cubic feet

3. Box 2: Volume = 16 sq feet x 3 feet = 48 cubic feet

4. Box 3: Volume = 16 sq feet x 5 feet = 80 cubic feet

5. Total Volume of three boxes = 48 + 48 + 80 = 144 cubic feet

The following statements are true about parallelograms: diagonals bisect each other, opposite sides are congruent, opposite angles are congruent and opposite angles are supplementary.

A quadrilateral with two sets of parallel sides is referred to as a parallelogram. In a parallelogram, the opposing sides are of equal length, and the opposing angles are of equal size. Additionally, the interior angles that are supplementary to the transversal on the same side. 360 degrees is the total of all interior angles.

A parallelepiped is a three-dimensional shape with parallelogram-shaped faces. The base (one of the parallel sides) and height (the distance from top to bottom) of the parallelogram determine its area. A parallelogram's perimeter is determined by the lengths of its four sides.

The properties of a parallelogram are shared by the shapes of a square and a rectangle.

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The textile factory does not have enough dye to dye 5 rolls of fabric using 4 buckets of dye which is certain feet long.

Calculating the amount of dye needed to color one roll of fabric and comparing it to the total amount of dye available will help us decide whether the textile mill can dye 5 rolls of cloth with 4 buckets of dye.

We are aware that a single roll of fabric is 1,067 feet long and need for a 43-liter dye bucket. We divide the dye's volume by the fabric's length to get how much dye is needed per foot of fabric:

0.0403 litres per foot or 43 litres divided by 1,067 feet

Consequently, the factory needs 0.0403 litres of dye to colour one foot of fabric.

Considering there are 4 buckets and each bucket holds 43 litres of colour, the total amount of dye accessible is:

4 buckets at a rate of 43 litres each equal 172 litres.

The factory must dye a total of: to dye 5 rolls of fabric.

5,335 feet are equal to 5 rolls at 1,067 feet each.

We calculate the required amount of dye per foot by the total length of fabric to see if the factory has enough dye to colour this length of fabric:

5,335 feet * 0.0403 litres per foot = 215.1 litres

The manufacturer cannot colour 5 rolls of fabric with 4 buckets of dye because the whole amount of dye needed (215.1 litres) is more than the total amount of dye available (172 litres).

In conclusion, there isn't enough dye in the textile plant to colour 5 rolls of fabric using 4 buckets of dye.

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To find the value of tan(-284°24'), we can use the fact that the tangent function has period π, which means that:

tan(x) = tan(x + nπ)

where n is any integer. We can use this fact to convert the angle -284°24' to an equivalent angle between 0° and 360°:

-284°24' = -360° + 75°36' = 75°36'

Now, we need to find the reference angle, which is the acute angle between the terminal side of the angle and the x-axis. Since 75°36' is in the second quadrant (where the tangent function is positive), the reference angle is:

75°36' - 180° = -104°24'

Finally, we can use the identity:

tan(-θ) = -tan(θ)

to find the value of tan(-104°24'):

tan(-104°24') = -tan(104°24')

We can use a calculator to find that:

tan(104°24') ≈ 2.3835

Therefore:

tan(-284°24') ≈ -2.3835 (rounded to the nearest ten-thousandth)

So, the value of tan(-284°24') to the nearest ten-thousandth is -2.3835.

Answer:32

Step-by-step explanation:

Answer:

The formula for the volume of a cylinder is:

V = πr^2h

where V is the volume, r is the radius, and h is the height.

Substituting the given values, we get:

V = π(7ft)^2(31ft)

V = π(49ft^2)(31ft)

V = 49π(31ft^3)

V ≈ 15,067.05 cubic feet (rounded to two decimal places)

Therefore, the volume of the silo is approximately 15,067.05 cubic feet.

According to the question, the total cost to complete the company taxes is $135.

We can use the following equation to describe the problem:

Total cost = Fixed cost + Hourly rate * Number of hours

In this case, the fixed cost is $100 to complete the company taxes, and the accountant charges an additional $35 per hour to complete the taxes. Let's say it takes the accountant "h" hours to complete the taxes. Plugging in these values, we get:

Total cost = $100 + $35/h * h

Simplifying the equation, we get:

Total cost = $100 + $35

Total cost = $135

Therefore, the total cost to complete the company taxes is $135.

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Answer:

[tex]sin( \alpha ) = \frac{12}{13} [/tex]

[tex] \cos( \alpha ) = \frac{5}{13} [/tex]

[tex] \tan( \alpha ) = \frac{12}{5} [/tex]

It follows that:

[tex] \cot( \alpha ) = \frac{5}{12} [/tex]

[tex] \sec( \alpha ) = \frac{13}{5} [/tex]

Answer:

C) y = 5x - 7

Step-by-step explanation:

Slope- intercept form: y =mx +b

Here, m is the slope and b is the y intercept.

    Substitute m = 5 and (-1 , -12) in the above equation and find the value of b.

         -12 = 5*(-1) + b

        -12 = -5 + b

   -12 + 5 = b

            b = -7

Equation in slope intercept form:

               [tex]\boxed{\bf y = 5x -7}[/tex]

Answer:

x = 2,5

Step-by-step explanation:

3(3 - 2x) = 2x - 11

Multiply every term inside the bracket by the term on the outside:

9 - 6x = 2x - 11

Collect like terms and add them together (also, when moving the terms to the other side, the sign changes to the opposite)

-6x - 2x = -11 - 9

-8x = -20 / : (-8)

x = 2,5

The function of the sequence is T(n) = n² + n and it is quadratic

The domain of the graph is [0, 18]

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

Step          Cells

1                 2

2                6

3                12

From the table, we have

T(n) = n * (n + 1)

This gives

T(n) = n² + n

Hence, the function is T(n) = n² + n

The sequence T(n) = n² + n is a quadratic sequence

This is the set of the x values

From the graph, we have

x = 0 to 18

So, we have

Domain = [0, 18]

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Answer:

based on the information provided, Madelyn has a total of 26 dogs in her animal rescue. Of these, there are 13 smaller dogs who consume a total of 19.5 cups of food per day (1.5 cups per dog), and there are 13 larger dogs who consume a total of 39 cups of food per day (3 cups per dog).

Step-by-step explanation:

We know that the total number of cups of food consumed by the dogs is 39. We can use this information to create an equation:

1.5s + 3l = 39

We also know that Madelyn has a total number of dogs, which we can represent as:

s + l = total number of dogs

However, we cannot solve for s and l with only one equation. We need another equation to help us solve for both variables.

From the information provided, we know that each smaller dog gets 1.5 cups of food per day and each larger dog gets 3 cups of food per day. Therefore, the total amount of food consumed in one day can also be represented as:

1.5s + 3l = total cups of food consumed in one day

Since we know that the total cups of food consumed in one day is equal to 39, we can substitute this value into the equation:

1.5s + 3l = 39

1.5s + 3l = 39

2s + 4l = 52

Now we have two equations with two variables:

s + l = total number of dogs

2s + 4l = 52

We can use substitution or elimination to solve for s and l. Using substitution:

s + l = total number of dogs

s = total number of dogs - l

2(total number of dogs - l) + 4l = 52

2total number of dogs - 2l + 4l = 52

2total number of dogs + 2l = 52

total number of dogs + l = 26

Now we have two equations:

s + l = total number of dogs

total number of dogs + l = 26

We can use substitution again:

s + l = total number of dogs

s = total number of dogs - l

(total number of dogs - l) + l = 26

total number of dogs = 26

Therefore, Madelyn has a total of 26 dogs. We can now solve for the number of smaller and larger dogs using one of the previous equations:

s + l = total number of dogs

s + (26 - s) = 26

s = 13

Therefore, Madelyn has 13 smaller dogs and 13 larger dogs.

Answer:

A) 4 x (one-half x 6) ÷ 3

Step-by-step explanation:

Hope it helps:)

The calculated areas are the same for regular polygon and equal to 43.5 units squared.

The area of polygon is given by the formula:

A = (1/2)ap,

where A is the area, an is the apothem (the distance from the centre of the polygon to the midpoint of any side), and p is the polygon's perimeter, is the formula for calculating the area of a regular polygon.

The apothem of the equilateral triangle can be found using the formula:

a = s / (2 tan(π/n))

Substituting the value of s = 10 we have, and n = 3 side.

a = 10/ / (2 tan(π/3))

a ≈ 2.9 units

The perimeter of the triangle is given as:

p = 3 x 10 = 30 units

Now, the area of the triangle is given as:

A = (1/2)ap

A = (1/2)(2.9)(30) = 43.5 sq. units.

Hence, the calculated areas are the same and equal to 43.5 units squared.

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Answer:

Step-by-step explanation:

Bianca’s answer.

The apothem, rounded to the nearest tenth, is

✔ 2.9 units.

The perimeter of the equilateral triangle is

✔ 30 units.

Therefore, the area of the equilateral triangle is

✔ 1/2(2.9)(30) , or approximately 43.5 units2.

The calculated areas are

✔ the same, despite using different formulas

.

(A)  the annual rate of change between 1992 and 2006 was 0.0804

(B) r = 0.0804 * 100% = 8.04%

(C) value in 2009 = $11,650

What is the rate of change?

The rate of change is a mathematical concept that measures how much one quantity changes with respect to a change in another quantity. It is the ratio of the change in the output value of a function to the change in the input value of the function. It describes how fast or slow a variable is changing over time or distance.

A) The initial value is $44,000 and the final value is $15,000. The time elapsed is 2006 - 1992 = 14 years.

Using the formula for an annual rate of change (r):

final value = initial value * [tex](1 - r)^t[/tex]

where t is the number of years and r is the annual rate of change expressed as a decimal.

Substituting the given values, we get:

$15,000 = $44,000 * (1 - r)¹⁴

Solving for r, we get:

r = 0.0804

So, the annual rate of change between 1992 and 2006 was 0.0804 or approximately 0.0804.

B) To express the rate of change in percentage form, we need to multiply by 100 and add a percent sign:

r = 0.0804 * 100% = 8.04%

C) Assuming the car value continues to drop by the same percentage, we can use the same formula as before to find the value in the year 2009. The time elapsed from 2006 to 2009 is 3 years.

Substituting the known values, we get:

value in 2009 = $15,000 * (1 - 0.0804)³

value in 2009 = $11,628.40

Rounding to the nearest $50, we get:

value in 2009 = $11,650

Hence, (A)  the annual rate of change between 1992 and 2006 was 0.0804

(B) r = 0.0804 * 100% = 8.04%

(C) value in 2009 = $11,650

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