Triangle ABC ~ triangle DEF.

triangle ABC with side AB labeled 11, side CA labeled 7.6 and side BC labeled 7.9 and a second triangle DEF with side DE labeled 2.2

Determine the measurement of FD.

FD = 1.52
FD = 1.58
FD = 1.1
FD = 5.5

The new profit-sharing ratio is as follows:

Mazibuko's share: Vogel's share = 5: 8

Let's begin by determining the share of profits that each member of the original partnership—Vogel and Mazibuko—receives :

Vogel's share 1/(1+3) = 1/4

Share of Mazibuko: 3/(1+3) = 3/4

Let's now evaluate the new alliance with Malikane. The remaining 5/6 of the profits are divided between Vogel and Mazibuko in accordance with their prior profit-sharing ratio because Malikane receives a 1/6 part of the profits.

As a result, Vogel and Mazibuko share in the following profits from the new partnership:

Share for Vogel: (1/4) * (5/6) = 5/24

Share for Mazibuko: (3/4) * (5/6) = 15/24 = 5/8

The new profit-sharing ratio is as follows:

Mazibuko's share: Vogel's share = 5: 8

So, Vogel obtains 5 of the profits from the new partnership's 13 shares, whereas Mazibuko receives 8 of the income from the 13 shares.

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The distributive pattern of 3 * 45 that can be expressed is:

3(40) + 3(5) = 135.

The Distributive Property is a fundamental property of arithmetic that explains how to simplify expressions that involve multiplication and addition or subtraction, for example:

a(b + c) can be expressed as a*b + a*c = ab + ac.

In the image given we have that 45 multiplied by 3 will give us 135. This can be written using the distributive pattern as:

3(45) = 3(40) + 3(5)

= 120 + 15

= 135

Therefore, 3(45) = 3(40) + 3(5)

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One possible rational function with the given characteristics is r(x) = 4 + (x + 9)(x + 2)(x - 4) / [(x - 3)(x + 5)].

In general, a function is a mathematical concept that defines a relationship between an input (or set of inputs) and an output (or set of outputs). It is a rule that takes an input value and produces a corresponding output value.

One possible rational function with the given characteristics is:

r(x) = 4 + (x + 9)(x + 2)(x - 4) / [(x - 3)(x + 5)]

Explanation:

The factor (x + 9) in the numerator makes the function cross the x-axis at x = -9.

The factor (x - 4) in the numerator makes the function touch the x-axis at x = 4.

The factors (x - 3) and (x + 5) in the denominator create vertical asymptotes at x = 3 and x = -5, respectively.

The factor (x + 2) in the numerator creates a hole at x = -2, since it cancels out the factor (x + 2) in the denominator.

Finally, the constant term 4 in the numerator and the absence of any higher degree terms in the numerator or denominator create a horizontal asymptote at y = 4.

Note that there may be other rational functions with the same characteristics, but this is one possible solution.

Therefore, One possible rational function with the given characteristics is r(x) = 4 + (x + 9)(x + 2)(x - 4) / [(x - 3)(x + 5)].

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Answer:

He can only wear 2 outfits because he only has 2 shirts.

Step-by-step explanation:

Hope it can help

Answer:

2 outfits because he only has 2 shirts

Step-by-step explanation:

Answer:

  about 17.7 mph

Step-by-step explanation:

You want the value of v that maximizes F(v) = 3600v/(22+0.07v²).

The derivative of F is ...

  [tex]F'(v)=\dfrac{3600(22+0.07v^2)-3600v(0.14v)}{(22+0.07v^2)^2}[/tex]

The derivative is zero at the maximum. This is the case when the numerator is zero:

  3600(22 +.07v² -.14v²) = 0

  .07v² = 22

  v = √(22/0.07) ≈ 17.728

Flow rate is maximized at a speed of about 17.7 miles per hour.

Answer:

................................

Step-by-step explanation:

............................................................

............................................................

Answer:

$4.32

Step-by-step explanation:

Cost Price: $48

Selling Price: $52.32

tax = S.P - C.P

tax = $52.32 - $48 = $4.32

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The value of the given expression is 16.

The expression we are interested in is:

logₐ  ((x²y)/(z³))

To simplify this expression, we can use the properties of logarithms. In particular, we can use the fact that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator. Similarly, the logarithm of a power is equal to the product of the logarithm of the base and the exponent.

Using these properties, we can rewrite the expression as follows:

log ₐ ((x²y)/(z³)) = log ₐ (x²) + log ₐ (y) - log ₐ (z³)

Now, we can substitute the given values of the logarithms of x, y, and z:

log ₐ (x²) = 2 log ₐ (x) = 2 * 5 = 10 log ₐ (y) = 3 log ₐ (z³) = 3 log ₐ (z) = 3 * (-1) = -3

Substituting these values back into the expression, we get:

log ₐ ((x²y)/(z³)) = 10 + 3 - (-3) = 16

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The components of each vector is :

u = (0,2)

v= (-4.9149, -3.4416)

Magnitude = 5.1

Direction = 196 deg

To find the magnitude and direction of the sum

.The magnitude and direction of the sum of two or more vectors can be calculated with vector addition.

To get the magnitude of the sum of two vectors (A and B), you can use the Pythagorean theorem: |C| = sqrt(A^2 + B^2 + 2ABcosθ).

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The volume of the family value box increases by approximately 44% compared to the regular box.

The volume of a rectangular box can be calculated as the product of its length, width, and height.

Let's assume the dimensions of the regular box are L, W, and H, and the dimensions of the family value box are 1.25L, W, and 1.15H, respectively.

The volume of the regular box is LWH, and the volume of the family value box is

(1.25L)(W)(1.15H) = 1.4375LWH.

To find the percentage increase in volume from the regular box to the family value box, we can use the following formula:

Percentage increase = (New value - Old value) ÷ Old value x 100%

In this case, the old value is the volume of the regular box, and the new value is the volume of the family value box. Therefore, the percentage increase in volume is:

(1.4375LWH - LWH) ÷ LWH x 100% ≈ 44%

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The correct question is:

A food company sells its cornflakes in boxes of two different sizes: the regular box and the family value box. For the family value box, the length of the box has been increased by 25% the height has been increased by 15% and the width remains the same. By what percentage does the volume of the box increase from the regular box to the value box? Round your answer to the nearest percent.

We can explain this  statement as describing an array of subsets of S whereby  each subset in the array  has a relationship to other subsets in a one way or another.

In mathematics, set A is a subset of a set B if all elements of A are also elements of B; B is then a superset of A.

\So we can say that a  proper subset is one that contains a few elements of the original set whereas an improper subset contains every element of the original set.

So in the scenario shown above, we can deduce that  for every pair of subsets Di and Dj in the array, there is another  third subset in the array  that contains both Di and Dj.

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The answers are :

a) The graph is translated upwards by 4 units.

b) The graph is translated downwards by 2 units.

Since, The modification of an existing graph or graphed equation to create a different version of the following graph is known as translation.

a) The graph of y = f(x) is shown.

The other function is y = f(x) + 4.

As it can be observed that the graph is translated from its parent function which is y = f(x) to function y = f(x) + 4.

The graph is shifted down by 4 units or translated upwards by 4 units.

b) Here , the graph of y = g(x) is shown.

The other function is y = g(x -2)

As we can observe that the graph is translated from its parent function which is y = g(x) to function y = g(x - 2).

The graph is shifted downwards by 2 units or translated downwards by 2 units.

Therefore , the answers are :

a) The graph is translated upwards by 4 units.

b) The graph is translated downwards by 2 units.

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The total selling price of the 300 toys bought at a cost of $24 per unit with a mark up of 25% is $9,000.

The mark up refers to the percentage added on the cost of a retail item.

The mark up increases the cost price to determine the selling price.

The difference between the selling price and the cost price is the profit margin.

The number of toys bought by the store = 300

The unit purchase cost = $24

The total cost = $7,200 (300 x $24)

The mark up = 25%

The selling price for 300 = $9,000 ($7,200 x 1.25)

The selling price per unit = $30 ($9,000 ÷ 300)

Thus, after marking up the items by 25%, the total selling price is $9,000.

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Therefore, the greatest amount of money Hayley can collect by selling the bread made with 22 cups of sugar and 4 sticks of butter is $154.

In mathematics, an equation is a statement of equality between two expressions, which are composed of variables, numbers, and mathematical operations. An equation typically has one or more variables that can take different values, and the goal is often to find the values of the variables that make the equation true. Equations can be used to model a wide range of real-world situations, from simple problems like calculating the area of a rectangle, to complex systems like those found in physics and engineering. Solving equations is a fundamental skill in mathematics, and there are many techniques and strategies for finding solutions to different types of equations.

Here,

To find the maximum amount of money Hayley can collect by selling bread, we need to find the maximum number of loaves of each type of bread she can make with 22 cups of sugar and 4 sticks of butter.

Let's assume Hayley needs x cups of sugar and y sticks of butter to make one loaf of zucchini bread, and a cups of sugar and b sticks of butter to make one loaf of banana bread. We have the following system of equations:

x + a = 22 (total cups of sugar)

y + b = 4 (total sticks of butter)

To maximize the amount of money Hayley can collect, she should make as many loaves of zucchini bread as possible, since it sells for a higher price. Therefore, we want to maximize the value of x.

Solving the system of equations, we get:

x = 22 - a

y = 4 - b

To maximize x, we want to minimize a. Let's assume a = 0 (i.e., Hayley uses all 22 cups of sugar for zucchini bread), then we get:

x = 22

y = 4 - b

To make one loaf of banana bread, Hayley needs b sticks of butter. Since b is a positive integer, the maximum value of y is 3 (when b = 1). Therefore, Hayley can make 3 loaves of banana bread and 22 loaves of zucchini bread.

The total amount of money Hayley can collect is:

3($3) + 22($4) = $66 + $88

= $154

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Complete question:

She can sell zucchini bread for $4 and banana bread for $3. What is the greatest amount of money Hayley can collect by selling the bread made with 22 cups of sugar and 4 sticks of butter?

The three cube roots of 4√3 + 4i are:

2(cos(π/18) + i sin(π/18))

2(cos(7π/18) + i sin(7π/18))

2(cos(11π/18) + i sin(11π/18))

To find the cube roots of a complex number, you can follow these steps:

Step 1: Write the complex number in polar form.

Step 2: Use the formula for finding the cube roots of a complex number in polar form. The formula for finding the cube roots of a complex number in polar form is:

        [tex]z^{\frac{1}{3}} = r^{\frac{1}{3} } [cos((\theta + 2k\pi)/3) + i sin((\theta + 2k\pi )/3)][/tex]

Step 3: Substitute the values into the formula.

Substitute the values for r and θ into the formula, and simplify.

Step 4: Calculate the cube roots.

Step 5: Write the cube roots in rectangular form.

Here we have 4√3 + 4i

The polar form of the complex number is

=> 4√3 + 4i = 8(cos(π/6) + i sin(π/6))

Using the formula, [tex]z^{\frac{1}{3}} = r^{\frac{1}{3} } [cos((\theta + 2k\pi)/3) + i sin((\theta + 2k\pi )/3)][/tex]

[tex]z^{\frac{1}{3}} = 8^{\frac{1}{3} } [cos((\pi /6 + 2k\pi)/3) + i sin((\pi/6 + 2k\pi )/3)][/tex]  

where k = 0, 1, 2.

Now, we can calculate the cube roots by substituting the values of k into the formula and simplifying. Here are the three cube roots:

Substituting k = 0, 1, 2 into the formula, we get:

k = 0: [tex]z^{1/3}[/tex] = 2(cos(π/18) + i sin(π/18))

k = 1:  [tex]z^{1/3}[/tex] = 2(cos(7π/18) + i sin(7π/18))

k = 2:  [tex]z^{1/3}[/tex]  = 2(cos(11π/18) + i sin(11π/18))

Therefore,

The three cube roots of 4√3 + 4i are:

2(cos(π/18) + i sin(π/18))

2(cos(7π/18) + i sin(7π/18))

2(cos(11π/18) + i sin(11π/18))

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In the given graphs, the first and third graphs are connected graphs.

A connected graph is a collection of vertices, or points, and edges, or lines, such that there is a path between every pair of vertices.

This means that it is not possible to find any isolated vertices or disjoint subsets of vertices, and every vertex is connected to at least one other vertex.

The edges of a connected graph represent the relationships or connections between the vertices.

Here we have 3 graphs

In the first graph, every vertex is connected to other vertices and we can go from one node to another node

Hence, The first graph is a connected graph

In the second graph, there are two graphs that are not connected to other. Hence, The Second graph is not a connected graph

In the third graph, every node is connected

Hence, The third graph is a connected graph.  

Therefore,

In the given graphs, the first and third graphs are connected graphs.

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ACF is a 30º-60º-90º triangle because of the following:

1) Based on a theorem, in a 30°-60°-90° triangle the sides are in the ratio 1 : 2 : [tex]\sqrt{3}[/tex]

1 → short leg

2 → hypotenuse

[tex]\sqrt{3}[/tex] → long leg

Side length of the hexagon is the short leg of the triangle. It is 1.

r1 is the radius of the incircle in a regular hexagon. 2(r1) is the diameter of the incircle. It is also the hypotenuse of the right triangle. It is 2.

Using Pythagorean theorem.

[tex]a^2 + b^2 = c^2[/tex]

[tex]1^2 + b^2 = 2^2[/tex]

[tex]b^2 = 2^2 - 1^2[/tex]

[tex]b^2 = 4 - 1[/tex]

[tex]b^2 = 3[/tex]

[tex]\sqrt{b^2} = \sqrt{3}[/tex]

[tex]b = \sqrt{3}[/tex]

Answer:

3x-y=5

Step-by-step explanation:

standard form is 3x-y=5

The total surface area of the rectangular prism is 144 square inches.

To find the total surface area of the rectangular prism, we need to find the area of all six faces and then add them together. We are given the dimensions of three of the rectangular faces, and we can assume that the other three faces have the same dimensions as the ones given.

The area of a rectangular face can be found by multiplying the length and width. Using the dimensions given in the net, we can find the area of each face:

The top and bottom faces both have dimensions of 8 inches by 3 inches, so their areas are:

A = 8 x 3 = 24 square inches (each)

The left and right faces both have dimensions of 6 inches by 4 inches, so their areas are:

A = 6 x 4 = 24 square inches (each)

The front and back faces both have dimensions of 8 inches by 6 inches, so their areas are:

A = 8 x 6 = 48 square inches (each)

To find the total surface area, we add up the areas of all six faces:

Total surface area = 2(24) + 2(48) + 2(24) = 144 square inches

Therefore, the total surface area of the rectangular prism is 144 square inches.

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The probability that Herman selects a fiction book is given as follows:

P(fiction) = 0.4.

A probability is calculated as the division of the desired number of outcomes by the total number of outcomes in the context of a problem/experiment.

The and/or probabilities are related as follows:

P(A and B) = P(A) + P(B) - P(A or B).

In the context of this problem, we have that:

P(hardcover and fiction) = P(hardcover) + P(fiction) - P(hardcover or fiction).

Hence the probability of selecting a fiction book is obtained as follows:

0.2 = 0.6 + P(fiction) - 0.8

P(fiction) = 0.4.

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A nurse is caring for a pediatric patient weighing 8.250 kilograms.

We will solve the problem of unit conversion. This time we will convert grams to kilograms as;

Recall that

[tex]{ \ 1 \ kilogram = 1,000 \ grams \ }[/tex]

8,250 grams = n kilograms

[tex]= 8,250 \grams \times \frac{1 \ kilogram}{1,000 \ grams} \ }\\= 8,250 \times 1 \ kilogram \\= 8.250 \ kilograms \ }}[/tex]

Thus, after the conversion process, the patient weight is 8.250 kilograms.

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Answer:

$585/month

Step-by-step explanation:

To determine how much money Mr. Smith has left over monthly to put into savings, we need to first calculate his monthly income and his monthly expenses, including taxes.

Mr. Smith's hourly rate is $20, and he works full time, which is typically 40 hours per week. Therefore, his weekly income is:

$20/hour x 40 hours/week = $800/week

To find his monthly income, we multiply his weekly income by the number of weeks in a month:

$800/week x 4 weeks/month = $3,200/month

To find his monthly expenses, we add up the amounts for his household and automobile expenses:

$1,410/month (household) + $405/month (automobile) = $1,815/month

Since Mr. Smith gets about 25% of his income taken out for taxes, we need to calculate the amount of taxes he pays each month:

$3,200/month x 0.25 = $800/month (taxes)

Now we can subtract his monthly expenses and taxes from his monthly income to find out how much money he has left over to put into savings:

$3,200/month - $1,815/month - $800/month = $585/month

Therefore, Mr. Smith has $585/month left over to put into savings after paying his expenses and taxes.

Im diying:(

dont be mean man

After 20 minutes the jello salad will have a temperature of 43.9 F.

We have,

Newton's Law of Cooling is given by the formula

T(t) = T(s) + (T(0) - T(s))[tex]e^{-kt[/tex]

So, T(t) = 68 + (50 - 68) [tex]e^{-(0.01457). 20[/tex]

T(t) = 68 + (-18) [tex]e^{-(0.2914)[/tex]

T(t) = 68 + (-18) (1.338299)

T(t) = 68 - 24.089382

T(t) = 43.910618

T(t) = 43.9F

Thus, After 20 minutes the jello salad will have a temperature of 43.9 F.

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The inequality 1 - 3x > 4( 2x - 6 ) is equivalent to x < 25 / 11

To solve the inequality 1 - 3x > 4( 2x - 6 ), we can start by simplifying the right-hand side:

4( 2x - 6 ) = 8x - 24

When we insert this into the original inequality, we get:

1 - 3x > 8x - 24

Adding 3x to both sides, we get:

1 > 11x - 24

Adding 24 to both sides, we get:

25 > 11x

Dividing both sides by 11, we get:

x < 25 / 11

Therefore, the inequality 1 - 3x > 4( 2x - 6 ) is equivalent to x < 25 / 11

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The total surface area of the cylinder is 62.8 ft². And the right option is D. 62.8.

A cylinder is a solid shape that is made up of a rolled surface with a circular top and a circular base.

To calculate the total surface area of the cylindrical drum, we use the formula below.

Formula:

Where:

From the question,

Given:

Substitute these values into equation 1

Hence the total surface area is 62.8 ft².

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Step-by-step explanation:

Each year the truck retains 85 % of its value   ( 85% + 15% = 100%)

45000 * .85 * .85 * .85 * .85   = $ 45 000 *  .85^4 = $ 23490.28

Answer:

The value of the truck would be $23,490.28 after 4 years.

Step-by-step explanation:

Main concepts:

Concept 1. Percentages

Concept 2. Depreciation

Concept 3. Distributive property

Concept 4. Repeated multiplication as a power

Concept 1. Percentages

A percentage is effectively a unit.  In the same way that 25¢ is the same as 0.25 US dollars, 15% is the same as 0.15 (unit-less number).

A common task is to find a percentage of a quantity.  For example, finding 25% of 40.  Mathematically, the word "of" here means multiply, so 25% of 40 means "25% * 40".  But, since a percentage can be converted to a regular unit-less number, this is also the same as "0.25 * 40" which is 10.  So, 25% of 40 is 10.

Concept 2. Depreciation

To depreciate means that the value of the thing decreases.  In this question, the value of the truck depreciates by 15% (per year), so its value would decrease by whatever 15% of its current value is.

15% of $45,000 is 0.15 * $45,000 = $6,750 (this is the amount that the value decreases by during the first year)

To find the value after 1 year, we need to subtract this from the original value.  We'll label this equation 1:

Common mistake: It is a common mistake for people to continue subtracting $6,750 for years 2, 3, and 4 thinking that this is subtracting another 15%.  However, each year, the 15% depreciation is based on the new value of the truck, so during the second year, the truck started at a value of $38,250 (from the end of the first year), and 15% of $38,250 will be different from 15% of $45,000.

15% of $38,250 is 0.15 * $38,250 = $5,737.50 (this is the amount that the value decreases by)

To find the value after 2 year, we need to subtract this from the value at the end of year 1.  We'll label this equation 2:

Concept 3. Distributive property

In Equation 1 above, the left side of the equation can be factored by using the distributive property in reverse, and factoring out "$45,000":

$45,000 - $6,750 is the same as

$45,000 * ( 1 - 0.15 ) = $38,250    -- Equation 1a

This expression illuminates what is happening to the value if we change the numbers inside the parentheses into percentages

$45,000 * ( 100% - 15% )

Notice that in the parentheses, we start with 100% (whatever 100% is), and we're subtracting 15%.

Similarly, in Equation 2, the left side of the equation can be factored by using the distributive property in reverse, and factoring out "$38,250":

$38,250 - $5,737.50 is the same as

$38,250 * ( 1 - 0.15 ) = $32,512.50   -- Equation 2a

Notice that in the parentheses, we again start with 100% (whatever 100% is), and we're subtracting 15%.

Concept 4. Repeated multiplication as a power

Quick recap from Equation 1a & Equation 2a:

For year 1, Equation 1a:  $45,000 * ( 1 - 0.15 ) = $38,250

For year 2, Equation 2a:  $38,250 * ( 1 - 0.15 ) =  $32,512.50

However, note the equivalence in bold above.  This means that we can replace the $38,250 in the "Year 2" equation with the entire left side of the equation from Year 1.

Thus, for Year 2, the equation become:

$45,000 * ( 1 - 0.15 ) * ( 1 - 0.15 ) =  $32,512.50

While the value at the end of year 3 could be written

$32,512.50 * ( 1 - 0.15 )

notice that the left side again starts with the value that ended the year before.  With another substitution, for Year 3, the equation becomes:

$45,000 * ( 1 - 0.15 ) * ( 1 - 0.15 ) * ( 1 - 0.15 ) =  Value at end of year 3

Continuing the pattern, we need the value at the end of the year before, times ( 1 - 0.15 ) to get the value for the end of the following year.  

$45,000 * ( 1 - 0.15 ) * ( 1 - 0.15 ) * ( 1 - 0.15 ) * ( 1 - 0.15 ) =  Value at end of year 4.

Notice that each year, we're multiplying by another factor of ( 1 - 0.15 )

Recall that multiplying by the same number or expression repeatedly can be simplified with an exponent.  For example, five 2s multiplied together: 2*2*2*2*2 can be written as [tex]2^5[/tex].  

Therefore, the value of the truck at the end of year 2 can be written as

[tex]\$45,000 * (1-0.15)^{2}[/tex]

The value of the truck at the end of year 3:  [tex]\$45,000 * (1-0.15)^{3}[/tex]

The value of the truck at the end of year 4:  [tex]\$45,000 * (1-0.15)^{4}[/tex]

Side note: the value of the truck at the end of any year "n" is [tex]\$45,000 * (1-0.15)^{n}[/tex]

So, to answer the question, and find the value of the truck in 4 years, we just need to evaluate [tex]\$45,000 * (1-0.15)^{4}[/tex]

[tex]\$45,000 * (1-0.15)^{4}=[/tex]

[tex]=\$45,000 * (0.85)^{4}[/tex]

[tex]=\$45,000 * 0.52200625[/tex]

[tex]=\$23,490.28125[/tex]

Rounded to the nearest penny, the value of the truck would be $23,490.28 after 4 years.

The exact values of sin(x/2), cos(x/2), and tan(x/2) are:

Sin (x/2) = √ [ 2 - √2)/4 ]

cos (x/2) = √[ (2 + √2)/4 ]

Tan (x/2) = √[ (2 + √2)]/ √[ (2 + √2)]

The tangent of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the adjacent side.

The tangent of an angle can be expressed in terms of the sine and cosine of the same angle using the identity: tan(x) = sin(x) / cos(x)

Here we have

tan x = 1

If tan(x) = 1, we can determine the values of sin(x) and cos(x) using the identity:

=> tan² (x) + 1 = sec² (x)

Substituting tan(x) = 1, we get:

1 + 1 = sec²(x)

2 = sec²(x)

sec(x) = √2

As we know sec(x) = 1/cos(x), so:

=> cos(x) = 1/sec(x) = 1/√2 = √2/2

Similarly, sin(x) = tan(x) × cos(x) = 1 × √2/2 = √2/2

Use the half-angle identities to find sin(x/2), cos(x/2), and tan(x/2):

=> sin(x/2) = ±√[(1 - cos(x))/2 ]

= ± √[(1 - √2/2)/2]

= ± √ [ 2 - √2)/4 ]

Since tan(x) = 1 is positive, we know that x is in the first quadrant, which means x/2 is also in the first quadrant.

Therefore, sin(x/2) is positive, so:

sin(x/2) = √ [ 2 - √2)/4 ]

cos(x/2) = ±√[ (1 + cos(x))/2]

= ±√ [ (1 + √2 /2)/2]  

= ± √[ (2 + √2)/4 ]

Since x/2 is in the first quadrant, cos(x/2) is also positive, so:

cos(x/2) = √[ (2 + √2)/4 ]

Finally, we can find tan(x/2) using the identity:

tan(x/2) = sin(x/2) / cos(x/2)

tan(x/2) = √[ (2 + √2)/4 ]/ √[ (2 + √2)/4 ]

tan (x/2) = √[ (2 + √2)]/ √[ (2 + √2)]

Therefore,

The exact values of sin(x/2), cos(x/2), and tan(x/2) are:

Sin (x/2) = √ [ 2 - √2)/4 ]

cos (x/2) = √[ (2 + √2)/4 ]

Tan (x/2) = √[ (2 + √2)]/ √[ (2 + √2)]

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Parent function: y = x (linear)

Equation of the graph: y = x - 4

      This function is a linear equation and the parent linear function is;

       y = x.

      These equations are written in the form y = mx + b where m is the slope and b is the y-intercept. Our y-intercept is -4 and our slope is 1 using the method "rise over run" where we count units up for every unit right. This means our equation is;

     y = x - 4

This prompt is about the a word problem involving an exponential function  given as  B(x) = 5 * 2²

a) The completed table is attached. See related comments there as well.

b) Graph is attached.

c)  It can be observed that this particular scenario exemplifies an exponential function, as the population of bacteria dwelling in Albert's body increases twofold every hour. Such a phenomenon indicates that the rate of growth is directly proportional to the current amount.

d) The function is given as: B(x) = 5 * 2²

e) Albert will become symptomatic when the amount of bacteria in his body reaches 1280. This will happen after 8 hours.

we can also solve this without using logarithms by repeatedly multiplying 256 by 1/2 until we get to 1, and counting the number of times we multiplied:

256 x  (1/2) = 128

128 x (1/2) = 64

64 x  (1/2) = 32

32 x (1/2) = 16

16 x (1/2) = 8

8 x   (1/2) = 4

4 x (1/2) = 2

2 x  (1/2) = 1

We multiplied  256 by 1/2 a  total of 8 times , so we know that

2⁸ = 256.

Therefore, we can write:    2ˣ = 256 as  2ˣ = 2⁸

Equating the exponents on both sides, we get:

x = 8

f) B(3) = 5 x 2³

B(3) = 5 x 8
B(3) = 40


Meaning: The above means that after 3 hours, the bacterial inAlbert's body will have reached 40. This is 8 times the initial amount.

G) To find B(5.5)

We place x with 5.5 in the above function

Thus, B(5.5) = 5 x 2^5.5
= 88.39

This means that after 5.5 hours, albert will have a bout 88.39 bacterial in his body.

H) This will take 9.97 hours

We need to find the value of x in the equation:

B(x) = 5 * 2^x = 5000

Dividing both sides by 5, we get:

2^x = 1000

Taking the logarithm of both sides to base 2, we get:

log2(2^x) = log2  (1000 )

Using the property that log2(a^b) =  b x log2 (a), we can simplify the left-hand side:

x * log2(2) = log2 (1000 )

Since log 2(2) = 1, we can simplify further to:

x = log 2(1000)

Using a calculator, we can evaluate this expression:

x ≈ 9.97


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