A pension fund manager decides to invest a total of at most $35 million in U.S. Treasury bonds paying 6% annual interest and in mutual funds paying 9% annual interest. He plans to invest at least $5 million in bonds and at least $20 million in mutual funds. Bonds have an initial fee of $100 per million dollars, while the fee for mutual funds is $200 per million. The fund manager is allowed to spend no more than $6000 on fees. How much should be invested in each to maximize annual interest? What is the maximum annual interest?
The amount that should be invested in Treasury bonds is $ 10 million and the amount that should be invested in mutual funds is $ 25 million.
The maximum annual interest is $

The vector AC can be expressed in terms of vector AB and vector BC as follows:

In Euclidean geometry, a parallelogram is described as a simple quadrilateral with two pairs of parallel sides.

We have that  ABCD is a parallelogram, vector AC is equivalent to vector BD, which can be expressed in terms of vector AB and vector BC as follows:

Vector BD = Vector AB + Vector BC

We can substitute BD for AC in the above equation because  vector AC is equivalent to vector BD, we obtain the following:

Vector AC = Vector AB + Vector BC

In conclusion, vector AC can be expressed in terms of vector AB and vector BC as the sum of vector AB and vector BC.

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The volume of the cylinder is 42080.87 cubic yd

Cylinder is a 3-dimensional solid shape that consists of two identical and parallel bases linked by a curved surface.It has two curved edges, one curved surface, and two flat faces.

Volume of a cylinder = π * r^2 * h

Where π = 3.14

r = 17.2 td

h = 45.3 yd

Volume of a cylinder = 3.14 * (17.2)^2 * 45.3

Volume = 3.14 * 295.84 * 45.3

Volume = 42080.87 cubic yard

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The sign of each of the functions in the given interval is positive.

Suppose \( \theta \) is in the interval \( \left(90^{\circ}, 180^{\circ}\right) \), we can find the sign of each of the following functions by using the unit circle and the reference angles.

77. \( \cos \frac{\theta}{2} \)
Since \( \theta \) is in the second quadrant, \( \frac{\theta}{2} \) will be in the first quadrant. Therefore, the sign of \( \cos \frac{\theta}{2} \) will be positive.

78. \( \sin \frac{\theta}{2} \)
Similarly, since \( \theta \) is in the second quadrant, \( \frac{\theta}{2} \) will be in the first quadrant. Therefore, the sign of \( \sin \frac{\theta}{2} \) will be positive.

79. \( \sec \frac{\theta}{2} \)
The secant function is the reciprocal of the cosine function, so the sign of \( \sec \frac{\theta}{2} \) will be the same as the sign of \( \cos \frac{\theta}{2} \), which is positive.

In conclusion, the sign of each of the functions in the given interval is positive.

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The maximum number of days she can drive the car is approximately 4 days.

A rental car company charges $61.79 per day to rent a car and $0.13 for every mile driven.

She plans to drive 50 miles. She has at most $230 to spend.

Therefore, the maximum number of days that Tallulah can rent the car while staying within her budget can be computed as follows:

Using equations,

y = 0.13a + 61.79b

where

Therefore, she has a budget of 230 dollars and she wants to ride for 50 miles.

230 = 0.13(50) + 61.79b

230 - 6.5 = 61.79b

223.5 = 61.79b

b = 223.5  / 61.79

b = 3.61709014404

Therefore,

b = 4 days

Hence, she can ride maximum of 4 days.

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Based on the given information, the cost of a pound of pistachios and a pound of cashews is $7.875 and $5.50, respectively.

To find the cost of a pound of pistachios and the cost of a pound of cashews, we can use a system of equations.

Let's let P represent the cost of a pound of pistachios and C represent the cost of a pound of cashews.

Then we can write two equations based on the information given:

4P + 1C = 37

3P + 3C = 48

Now we can use the elimination method to solve for one of the variables.

Let's multiply the first equation by -3 to eliminate the P variable:

-12P - 3C = -111

3P + 3C = 48

Adding these two equations together gives us:

-8P + 0C = -63

Simplifying gives us:

P = 63/8

This means that the cost for a pound of pistachios is $63/8 or $7.875.

Substitute the value of P to any of the two equations so we can have the value of C.

4(63/8) + 1C = 37

C = 11/2 = 5.5

Therefore, the cost of a pound of pistachios is $7.875 while the cost of a pound of cashews is $5.50.

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Answer:

An exponential function in the form of y = a(m) x is of the form:

y = a * m^x

where "a" is a constant, "m" is a positive constant (the base of the exponential), and "x" is the independent variable.

To determine the exponential function in this form, you need to have some information about the function, such as its value at a particular point or its rate of growth.

Here are the general steps to determine the exponential function in the form of y = a(m) x:

Determine the value of "a" by plugging in the known value of "y" and "x" into the equation.

Determine the value of "m" by solving for it using the given information.

For example, let's say you are given that the function passes through the point (2, 8) and has a growth rate of 3. To determine the exponential function in the form of y = a(m) x, you would follow these steps:

Plug in the values of x and y into the equation:

8 = a * m^2

Solve for "a" by isolating it on one side of the equation:

a = 8 / m^2

Use the given growth rate of 3 to find the value of "m":

m = 1 + r = 1 + 0.03 = 1.03

Plug in the value of "m" and the value of "a" you found in step 2 into the equation:

y = (8 / 1.03^2) * 1.03^x

Simplifying this equation gives:

y = 7.514 * 1.03^x

So the exponential function in the form of y = a(m) x that passes through the point (2, 8) and has a growth rate of 3 is y = 7.514 * 1.03^x.

Step-by-step explanation:

here is a more detailed explanation of the steps involved in determining an exponential function in the form of y = a(m) x:

Determine the value of "a" by plugging in the known value of "y" and "x" into the equation.

In the equation y = a(m) x, "a" is a constant that represents the value of y when x is equal to 0. So to find the value of "a", we need to know the value of y for a specific value of x, other than 0.

The graph of the equation x^2/49 + (y + 1)^2/4 = 1 is added as an attachment

The expression that represents the function in words is given as

StartFraction x squared Over 49 EndFraction + StartFraction (y + 1) squared Over 4 EndFraction = 1

Express the equation properly

So, we have

x^2/49 + (y + 1)^2/4 = 1

The above equation is an ellipse

Next, we plot the graph on a coordinate plane using a graphing tool

See attachment for the graph of the equation

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The 6th term of the sequence is -97.65625

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

A(n) = -2.5^(n-1)

In the sixth term, we have

n = 6

Substitute the known values in the above equation, so, we have the following representation

A(6) = -2.5^(6-1)

Evaluate

A(6) = -97.65625

Hence, the 6th term is -97.65625

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Step-by-step explanation:

when 2 shapes are similar, then that means (among few other things) that both have the same angles at their vertexes.

by looking at the relative sizes of the sides, we can map which vertex of one correlates to which vertex (and angle) in the other.

D -> X

C -> Y

G -> U

F -> V

E -> W

so, the angle W = angle E = 136°.

Answer:

Assuming that the expression is meant to be written as:

(a^5 ⋅ 2^3)^3

We can simplify it using the rules of exponents. First, we can simplify the expression inside the parentheses:

a^5 ⋅ 2^3 = a^5 ⋅ 8 = 8a^5

Now we can substitute this simplified expression into the original expression:

(8a^5)^3

To simplify this, we use the power rule of exponents, which states that when we raise a power to another power, we multiply the exponents:

(8a^5)^3 = 8^3 ⋅ (a^5)^3 = 512a^15

Therefore, the simplified expression is 512a^15.

To find the function values, we need to substitute the given values of x into the function:

a) f(2) = 95(0.84)^2 ≈ 66.96

This means that after 2 hours, there are approximately 66.96 milligrams of medicine remaining in the person's bloodstream.

b) f(6) = 95(0.84)^6 ≈ 35.33

This means that after 6 hours, there are approximately 35.33 milligrams of medicine remaining in the person's bloodstream.

To determine an appropriate domain for the function, we need to consider the context of the problem. The function represents the amount of medicine remaining in a person's bloodstream after taking a dose of medication, so the domain should be the set of non-negative real numbers, since the amount of medicine cannot be negative and time cannot be negative. Therefore, an appropriate domain for the function is [0, ∞).

Interpretation: The function f(x) = 95(0.84)^x models the amount of medicine, in milligrams, remaining in a person's bloodstream after x hours of taking the medication. For example, after 2 hours, there are approximately 66.96 milligrams of medicine remaining in the person's bloodstream.

After taking a dose of medication, the amount of medicine remaining in a person's bloodstream, in milligrams, after x hours can be modeled by the function

To find:

Interpret the given function values and determine an appropriate domain for the function.

Solution:

The general form of an exponential function is

Where, a is the initial value, 0<b<1 is decay factor and b>1 is growth factor.

We have,

Here, 110 is the initial value and 0.83 is the decay factor.

It means, the amount of medicine in the person's bloodstream after taking the dose is 110 milligrams and the amount of medicine decreasing in the person's bloodstream with the decay factor 0.83 or decreasing at the rate of (1-0.83)=0.17=17%.

We know that an exponential function is defined for all real values of x but the time cannot be negative. So, x must be non negative.

We know that  for any value of x. So,  for all values of x.

Therefore, domain of the function is  and the range is .

Hope this helps!!!

GTPex-

Answer:

Mai is correct. We can use exponential notation to represent the number of coins each day. Let's call the number of coins on the first day "1". Then the number of coins on each subsequent day is twice the number of coins on the previous day. So we have:

Day 1: 1

Day 2: 2 = 2^1

Day 3: 4 = 2^2

Day 4: 8 = 2^3

...

Day n: 2^(n-1)

To find the number of coins Mai has on the 6th day, we substitute n = 6 into the formula for the number of coins:

Day 6: 2^(6-1) = 2^5 = 32

So Mai has 32 coins on the 6th day. Writing out the product of 2's (1.2.2.2.2.2.2) is equivalent to writing 2^6 = 32.

To find out how many coins Mai has after 28 days, we substitute n = 28 into the formula for the number of coins:

Day 28: 2^(28-1) = 2^27 = 134,217,728

So after 28 days, Mai has 134,217,728 coins.

an = 2*an-1 + an-2 is the sequence's recursive rule, and the seventh term is 268.

A formula that explains how to move from one word in a sequence to the next is known as a recursive rule for a sequence.

The variable typically serves as a symbol for the concept of a number.

In other words, takes on the values 1, 2, 3, etc. for the first, second, third, etc. terms.

A recursive sequence is one in which terms are defined by reference to one or more previously defined terms.

The recursive formula an+1=an+d can be used to find the (n+1)th term of an arithmetic series if you know the nth term and the common difference, d.

So, we know that:
2 = 2 * 1 + 0

3 = 2 * 1 + 1

8 = 2 * 3 + 2

19 = 2 * 8 + 3

46 = 2 * 19 + 8

The sequence's nth term is as follows after the preceding sequence:

a7 = 2*an-1 + a7 - 2

For, a7:
a7 = 2*a7-1+a7-2

a7 = 2*a6+a5

a7 = 2*a6+46

a6 = 2*a5+a4

a7 = 2*(2*46+19)+46

a7 = 268

Therefore, an = 2*an-1 + an-2 is the sequence's recursive rule, and the seventh term is 268.

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If f(x) is the parent function and g(x) is the transformation of f(x), then the required functions are as follows:

5. g(x) = -4ˣ and 6. g(x) = -(1/5)ˣ.

A point is transformed when it is moved from where it was originally to a new location. Translation, rotation, reflection, and dilation are examples of different transformations.

5.

The given function is as follows:

f(x) = 4ˣ

When the function f(x) = 4ˣ is reflected about the x-axis, then we get the graph of function g(x).

So, g(x) = -4ˣ

6.

The given function is as follows:

f(x) = (1/5)ˣ

When the function f(x) = (1/5)ˣ is reflected about the x-axis, then we get the graph of function g(x).

So, g(x) = -(1/5)ˣ

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a) The exponential growth equation is y = ( 1.2 )ˣ

b) The exponential decay equation is y = ( 0.71 )ˣ

The exponential growth or decay formula is given by

x ( t ) = x₀ × ( 1 + r )ⁿ

x ( t ) is the value at time t

x₀ is the initial value at time t = 0.

r is the growth rate when r>0 or decay rate when r<0, in percent

t is the time in discrete intervals and selected time units

Given data ,

Let the exponential growth equation be represented as A

Now , the value of A is

y = ( 1.2 )ˣ

where the growth rate is r = 20 %

Let the exponential decay equation be represented as B

Now , the value of B is

y = ( 0.71 )ˣ

where the decay rate is r = 29 %

The graph A represents an exponential decay graph and the graph B represents an exponential growth graph

Hence , the exponential equations are solved

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The answer of possible rational zeros for the given function f(x)=-5x^(4)-4x^(3)+3x^(2)+5x+10 are ±1, ±2, ±5, ±10, ±1/5, and ±2/5

The Rational Zero Theorem states that if f(x) is a polynomial with integer coefficients, then any rational zero of f(x) must be in the form of p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.

To find the possible rational zeros for the given function f(x)=-5x^(4)-4x^(3)+3x^(2)+5x+10, we need to list the factors of the constant term (10) and the factors of the leading coefficient (-5).

Factors of 10: ±1, ±2, ±5, ±10
Factors of -5: ±1, ±5

Now, we can list all possible rational zeros by taking each factor of the constant term and dividing it by each factor of the leading coefficient:

Possible rational zeros: ±1/1, ±2/1, ±5/1, ±10/1, ±1/5, ±2/5, ±5/5, ±10/5

Simplifying the fractions gives us the final list of possible rational zeros:

Possible rational zeros: ±1, ±2, ±5, ±10, ±1/5, ±2/5, ±1, ±2

Removing duplicates, we get:

Possible rational zeros: ±1, ±2, ±5, ±10, ±1/5, ±2/5

Therefore, the possible rational zeros for the given function f(x)=-5x^(4)-4x^(3)+3x^(2)+5x+10 are ±1, ±2, ±5, ±10, ±1/5, and ±2/5.

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The probability of rolling a triangle is 9 out of 15, or 60%.

Its essential vision is that an individual will nearly certainly happen.

Then the probability is given as,

P = (Favorable event) / (Total event)

As per the given data:

Total sides is a die = 15

triangles = 9

circles = 4

squares = 2

The probability of rolling a triangle is 9 out of 15, or:

For probability in % = (Favorable outcomes/Total outcomes) × 100

= (9/15) × 100

= 60%

Hence, The probability of rolling a triangle is 9 out of 15, or 60%.

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The requried measure of the PQ in the triangle PQT is 16 in.

Similar triangles are those triangles that have similar properties,i.e. angles and proportionality of sides.

Here,
A figure of triangles has been shown, ΔPQT and ΔORS.
We have to determine the measure of side PQ.

Since the given triangle is similar to one another, we can apply the property of proportionality between the similar triangles. By which:

PR/PQ = RS/QT
PQ + 4 / PQ = 10/8
8PQ + 32 = 10PQ
2 PQ = 32
PQ = 16 in

Thus, the requried measure of the PQ in the triangle PQT is 16 in.

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The probability that the height of a randomly selected tree is as tall as mine or shorter is 0.1151. The probability that the full height of a randomly selected tree is at least as tall as hers is 0.0362.

To find the probability that the height of a randomly selected tree is as tall as yours or shorter, we need to find the z-score for your tree's height and use the standard normal table to find the corresponding probability. The z-score is calculated as follows:

z = (x - μ) / σ

where x is the observed value, μ is the mean, and σ is the standard deviation.

For your tree, the z-score is:

z = (44.7 - 56.6) / 9.9 = -1.198

Using the standard normal table, the probability that the height of a randomly selected tree is as tall as yours or shorter is 0.1151.

For your neighbor's tree, the z-score is:

z = (74.4 - 56.6) / 9.9 = 1.798

Using the standard normal table, the probability that the height of a randomly selected tree is at least as tall as hers is 1 - 0.9638 = 0.0362.

So, the probability that the height of a randomly selected tree is as tall as yours or shorter is 0.1151, and the probability that the height of a randomly selected tree is at least as tall as your neighbor's is 0.0362.

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Step-by-step explanation:

remember, sine is the up/down leg, cosine is the left/right leg.

the Hypotenuse is the radius of the circle around the trigonometric triangle.

so,

sin(R) = 5/13

cos(R) = 12/13

tan(R) = sin(R)/cos(R) = 5/13 / 12/13 = (5×13)/(13×12) =

= 5/12

Answer: (6x²+5) (6x²- 5)

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Step-by-step explanation:

The completely factored expression by pulling out the 15x^(7)+3x^(4)+21     is  3(5x^(7)+x^(4)+7).

A factored expression is an algebraic expression that has been broken down into its prime factors. This is usually done to simplify the expression and make it easier to solve for a particular variable or to factor out a common factor from multiple terms.

To factor the given expression, we need to find the greatest common factor (GCF) of the terms and then divide each term by the GCF to get the simplified expression.

The GCF of 15x^(7), 3x^(4), and 21 is 3. So, we can factor out 3 from each term to get the simplified expression:

15x^(7)+3x^(4)+21 = 3(5x^(7)+x^(4)+7)

Therefore, the completely factored expression is 3(5x^(7)+x^(4)+7).

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The perimeter of a rectangle is the sum of the lengths of all four sides, which can be expressed as:

P = 2x + 2y

where x is the length of the rectangle in centimeters and y is the width of the rectangle in centimeters.

If the perimeter of the rectangle is z, then we can write the formula as:

z = 2x + 2y

Simplifying this equation, we get:

p = z

Therefore, the formula for p, the perimeter of the rectangle, is:

p = 2x + 2y

where x is the length of the rectangle in centimeters and y is the width of the rectangle in centimeters.

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Answer:

Step-by-step explanation:

Answer:Price of the table yesterday is x.

Price of the table today is $513.

$513 is 76% of yesterday's price x. 76% can be also written down as 76 divided by 100 (percentage) or 76/100 = 0.76.

Formula we can set up now is: x= 513 / 0.76 (To get the price of yersterday's table we need to divide the price of today's table with percentage in decimal form).

x= 675

The price of yesterday's table was $675 since 76% of that price (675 times 0.76) equals 513 dollars.

Step-by-step explanation:

The simplified trigonometric expression is sin(a).

The functions of an angle in a triangle are known as trigonometric functions, commonly referred to as circular functions. In other words, these trig functions provide the relationship between a triangle's angles and sides. There are five fundamental trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant.

Here the given trigonometric expression is,

=> [tex]\frac{sec(a)}{cot(a)+tan(a)}[/tex]

We know that [tex]cot(a)=\frac{cos(a)}{sin(a)}[/tex] and [tex]tan(a)=\frac{sin(a)}{cos(a)}[/tex] and [tex]sec(a)=\frac{1}{cos(a)}[/tex]

Then,

=> [tex]\frac{\frac{1}{cos(a)} }{\frac{cos(a)}{sin(a)}+\frac{sin(a)}{cos(a)} }[/tex]

=> [tex]\frac{\frac{1}{cos(a)} }{\frac{cos^2a+sin^2a}{cos(a)sin(a)} }[/tex]

=> [tex]\frac{\frac{1}{cos(a)} }{\frac{1}{cos(a)sin(a)} }[/tex]

=> [tex]\frac{1}{cos(a)}\times\frac{sin(a)cos(a)}{1}[/tex]

=> sin a

Hence the simplified trigonometric expression is sin(a).

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    The angles of a triangle equal 180 degrees when added together. We will use this logic to solve all three for x. We will use inverse operations and combine like terms to solve.

[30] Answer: x = 36

    Step-by-step explanation:

2x + 2x + x = 180

5x = 180

x = 36

[31] Answer: x = 9

    Step-by-step explanation:

* the little box represents 90 degrees

7x + 3x + 90 = 180

10x = 90

x = 9

[32] Answer: x = 45

    Step-by-step explanation:

2x + x + x = 180

4x = 180

x = 45

The amount of dry cells that are taken from the full container is 473.

To find the total number of dry cells taken from the full container, we need to add up the number of dry cells placed in the stockroom, on the shelf, and sold to customers. We can use the following equation:

  Total dry cells taken = Dry cells in stockroom + Dry cells on shelf + Dry cells sold to customers

Plugging in the given values, we get:

  Total dry cells taken = 325 + 45 + 18 + 25 + 30 + 24 + 6

Using the order of operations, we can simplify the equation:

Total dry cells taken = 325 + 45 + 18 + 25 + 30 + 24 + 6
Total dry cells taken = 370 + 18 + 25 + 30 + 24 + 6
Total dry cells taken = 388 + 25 + 30 + 24 + 6
Total dry cells taken = 413 + 30 + 24 + 6
Total dry cells taken = 443 + 24 + 6
Total dry cells taken = 467 + 6
Total dry cells taken = 473

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Answer:

To solve the inequality:

-10 - 66h ≥ -3

We can begin by isolating the variable h on one side of the inequality. First, we can add 10 to both sides:

-10 + 10 - 66h ≥ -3 + 10

Simplifying, we get:

-66h ≥ 7

Next, we can divide both sides by -66, remembering to reverse the inequality since we are dividing by a negative number:

h ≤ -7/66

Therefore, the solution to the inequality is h ≤ -7/66.

Step-by-step explanation:

sin(30°) = ½

Sin(30°) have the value of √1/2 which gives ½