triangle DEF is rotated 90 to the right to format the triangle with the side lengths shown below

The amount that needs to be deposited to have a saving of $50,000 in 30 years at the given interest rate is $8,302.10.

The compound interest formula is expressed as;

P = A / (1 + r/n)^nt

Where P is principal, A is amount accrued, r is interest rate is compound period and t is time elapsed.

Given the data in the question;

First, convert rate from percent to decimal.

Rate r = 6%

Rate r = 6/100

Rate r = 0.06 per year

To determine the principal, plug the given values into the formula above and solve or P

P = A / (1 + r/n)^nt

P = $50,000 / (1 + 0.06/12)^( 12 × 30 )

P = $50,000 / (1 + 0.05)³⁶⁰

P = $50,000 / (1.05)³⁶⁰

P = $8,302.10

Therefore, the principal investment is $8,302.10.

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Given a Geometric Series:

Where "r" is the ratio.

By definition:

Answer:

  n < -15/4

Step-by-step explanation:

You want to use the discriminant to find the values of n for which the quadratic 3z² -9z = (n -3) has only complex solutions.

The discriminant of quadratic equation ax²+bx+c = 0 is ...

  d = b² -4ac

The given quadratic can be put in this form by subtracting (n-3):

  3z² -9z -(n -3) = 0

This gives us ...

and the discriminant is ...

  d = (-9)² -4(3)(-(n-3)) = 81 +12(n -3)

  d = 12n +45

The equation will have only complex solutions when the discriminant is negative:

  d < 0

  12n +45 < 0 . . . . . use the value of the discriminant

  n +45/12 < 0 . . . . . divide by 12

  n < -15/4 . . . . . . . subtract 15/4

There will be two complex solutions when n < -15/4.

Randy's initial money = $12

Randy's earnings per week = $6

Becky's initial money = -$6 (she owes )

Becky's earnings per week = $7

Number of weeks: x

The equation for each:

• Randy:

50 = 12 + 6x

• Becky:

50 = -6 + 7x

Solve each for x:

Randy:

50= 12 + 6x

50-12 = 6x

38 = 6x

38/6=x

x= 6.3

Becky:

50= -8 + 7x

50+8 =7x

58=7x

58/7=x

x= 8.28

Randy will take 6.33 weeks and Becky 8.28 weeks.

Answer:

A. Randy

Answer:

All of em, except 5

Step-by-step explanation:

5544 / 2 = 2772

5544 / 3 = 1848

5544 / 4 = 1386

5544 / 6 = 924

5544 / 7 = 792

5544 / 8 = 693

5544 / 9 = 616

The statement is false, the number can be rewritten as:

0.33... = 3/9

So it is a rational number, not irrational

Here we have the statement:

"the number 0.3333... repeats forever; therefore, its irrational"

This is false, and let's prove that.

our number is:

0.33...

Such that the "3" keeps repeating infinitely.

If we multiply our number by 10, we get:

10*0.33... = 3.33...

If we subtract the original number we get:

10*0.33... - 0.33... = 3

9*0.33... = 3

Solving that for our number we get:

0.33... = 3/9

So that number can be written as a quotient between two integers, which means that it is a rational number.

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The expression in 4a) is given below

Collecting similar terms using the division rule of indices, we will have

The above expression therefore becomes

Hence,

The final answer = x/3

The solution for the given system of equations is x[1] = -1 and x[2] = 2.

A system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables.

Given are the following equations as -

5 x[1] + 5 x[2] = 5

2 x[1] + 3 x[2] = 4

Assume that -

x[1] = a    

x[2] = b

Then, we can write the equations as -

5a + 5b = 5

2a + 3b = 4

Now -

5a + 5b = 5

5(a + b) = 5

a + b = 1

a = 1 - b

So, we can write -

2a + 3b = 4

as

2(1 - b) + 3b =4

2 - 2b + 3b = 4

b = 4 - 2

b = 2 = x[2]

Then

a = 1 - 2

a = -1 = x[1]

Therefore, the solution for the given system of equations is x[1] = -1 and x[2] = 2.

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The time on which parents evening was due to finish was 19 hour 20 minutes.

Time is the ongoing pattern of existence and things that happen in what seems to be an irreversible order from the past, through the present, and into the future.

It is a component quantity of various measurements used to order events, compare the length of events or the time gaps between them, and quantify rates of change of quantities in objective reality or in conscious experience. Along with the three spatial dimensions, time is frequently considered a fourth dimension.

The International System of Units is built upon the seven base units of measurement stipulated by the Système International d'Unités (SI), from which all other SI units are derived. The primary unit of time is the second. The second can be shortened using either the letter S or the letter sec.

20 consecutive appointments of 10 mins = 20 × 10 mins

                                                                     = 200 min

                                                                      = 3 hours 20 mins

A break of 15 mins = 3 hours 20 mins + 15 min

                               = 3 hours 35 mins

The time that the parents evening due to finish = 15h 45 min +3h 35 mins

                                                                               = 19h 20min

Thus, the time on which parents evening was due to finish was 19h 20min.

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ANSWER

a = √21

EXPLANATION

This is a right triangle, so we have to apply the Pythagorean Theorem to find the value of a.

We know the length of the hypotenuse which is 5, and the length of one of the legs, which is 2. The Pythagorean Theorem for this problem is,

Subtract 2² from both sides,

And take the square root to both sides,

Hence, the value of a is √21.

We have the following:

old salary: $88441

new salary: $96402

The year is approximately 52 weeks, therefore:

a. old salary for week:

b. new salary for week:

c. weekly raise

Answer:

1,017.36ft^2

Explanation:

Area of the circular pool = \pi r^2

r is the radius of the pool

Given

r = d/2

r = 36/2

r = 18ft

Area of the circular pool = 3.14(18)^2

Area of the circular pool = 3.14 * 324

Area of the circular pool = 1,017.36ft^2

By taking the quotient between the total volume and the number of hours, we conclude that she drinks 8 ounces per hour.

We know that Ziba has 4 bottles, each one with 6 ounces, so the total volume of water is:

V = 4*6 ounces = 24 ounces.

We know that she drinks that in 3 hours, so the amount that she drinks each hour is:

24 oz/3 = 8 oz

She drinks 8 ounces of water each hour.

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Thus, it is clear that the lines RS and TV in the preceding diagram are parallel to each other

If you extend a set of lines indefinitely, they will remain parallel and never cross each other even though they are on the same plane. The symbol || represents the collection of parallel lines. All parallel lines are always equally spaced apart. Investigate the characteristics of parallel lines.

When two lines in a plane are stretched infinitely in both directions and do not cross, they are said to be parallel.

To solve the given question we know,

angle 1= angle 2 and lines RV // TS

angle 4= angle 3(interior angles on parallel lines are equal)

angle 1=angle 4 (vertically opposite angles are equal )

angle 1= angle 3 (angle 4=angle 1)

angle 4=angle 2( angle 1=angle 4)

Now we can see that the sum of base angles of the diagram will be 180 because

180-angle 3= angle STV

angle 4=angleSTV+180 (angle 3=angle4)

we proved that the diagram is a parallelogram because base angles of the same side are supplementary:

Therefore , we can conclude that the lines RS // TV in the preceding diagram .

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The absolute value function can intersect a horizontal x-axis at zero, one, as well as two points.

Thus, depending on the way the graph has indeed been shifted and reflected, it could or might not intersect the horizontal axis.

The absolute value function can intersect the x-axis at zero, one, or two points.

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In inequality 20 >= 4/5 w, w is 25 or any real no. lower than< 25 and in inequality -8<-1/4m, m = any real no. greater than> 24 is are the solution.

An inequality compares two values and indicates whether one is lower, higher, or simply not equal to the other.

A B declares that a B is not equal.

When a and b are equal, an is less than b.

If a > b, then an is bigger than b.

(those two are called strict inequality)

The phrase "a b" denotes that an is less than or equal to b.

The phrase "a > b" denotes that an is greater than or equal to b.

We have give the inequality to solve

20 = 4/5w

w = 20 × 5/4

   = 25 or any real no. lower than< 25

Let -8 = -1/4m

m = -8 × -4

m = 24

so m = any real no. greater than> 24

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Answer: Total Length of ribbon is 99 cm

Step-by-step explanation:

Here ribbon was cut into three parts in the ratio 1:3:5

let x be the common multiple of the above ratio

therefore, the lengths of the three parts of the ribbon is 1x,3x,5x

now, given is that the shortest part i.e 1x is equals to 11cm

i.e   1x=11

        x=[tex]\frac{11}{1}[/tex]=11cm

     now lengths of the ribbon will be

      1x=11cm, 3x=3*11=33cm, 5x=5*11=55cm

 now total length of piece of ribbon = 1x+3x+5x=9x=9*11=99cm

Given:

The test claims that night students' mean GPA is significantly different from the mean GPA of day students.

Null hypothesis: the population parameter is equal to a hypothesized value.

Alternative hypothesis: it is the claim about the population that is contradictory to the null hypothesis.

For the given situation,

Null and alternative hypothesis is,

Answer: option f)

Answer:

(3, 0)

Maximum Value of Objective Function = 15

Step-by-step explanation:

This is a problem related to Linear Programming(LP)

In linear programming, the objective is to maximize or minimize an objective function subject to a set of constraints.

For example, you may wish to maximize your profits from a mix of production of two or more products subject to resource constraints.

Or, you may wish to minimize cost of production of those products subject to resource constraints..

The given LP problem can be stated in standard form as

Max 5x - 4y

s.t.

-x + 3 ≥ y    

y ≤ 0.5x  + 1  

x ≥ 0, y ≥ 0

The last two constraints always apply to LP problems which means the decision variables x and y cannot be negative

It is standard to express these constraints with the decision variables on the LHS and the constant on the RHS

Rewriting the above LP problem using standard notation,

Let's rewrite the constraints using the standard form:
- x + 3 ≥ y  
→  -x - y ≥ -3  
→ x + y ≤ 3   [1]

y ≤ 0.5x + 1

→ -0.5x + y ≤ 1  [2]

The LP problem becomes
        Max 5x - 4y

s. t.

        x + y ≤ 3            [1]
        -0.5x + y ≤ 1      [2]
        x ≥ 0                 [3]
        y ≥0                  [4]

With an LP problem of more than 2 variables, we can use a process known as the Simplex Method to solve the problem

In the case of 2 variables, it is possible to solve analytically or graphically. The graphical process is more understandable so I will use the graphical method to arrive at the solution

The feasible region is the region that satisfies all four constraints shown.

The graph with the four constraint line equations is attached. The feasible region is the dark shaded area ABCD

The feasible region has 4 corner points(A, B,C, D) whose coordinates can be computed by converting each of the inequalities to equalities and solving for each pair of equations.

It can be proved mathematically that the maximum of the objective function occurs at one of the corner points.

Looking at [1] and [2] we get the equalities
     x + y = 3        [3]
-0.5x + y = 1        [4]

Solving this pair of equations gives x = 4/3 and y = 5/3 or (4/3, 5/3)

Solving y = 0 and x + y = 3 gives point x = 3, y =0  (3,0)

The other points are solved similarly, I will leave it up to you to solve them

The four corner  points are
A(0,0)
B(0,1)
C(4/3, 5/3)
D(3,0)

The objective function is 5x - 4y

To find the values of x and y that maximize the objective function,

plug in each of the x, y values of the corner points

Ignoring A(0,0)

we get the values of the objective function at the corner points as

For B(0,1) => 5(0) - 4(1) = -4

For C(4/3, 5/3) => 5(4/3) - 4(5/3) = 20/3 - 20/3 = 0

For D(3, 0) => 5(3) - 4(0) = 15

So the values of x and y which maximize the objective function are x = 3 and y = 0 or point D(3,0)

The expression equivalent to  -x² - 36 is the one in option C.

(-x - 6i)*(x - 6i)

We can rewrite the given expression as:

-x² - 36 = -x² - 6²

And remember that the product of a complex number z = (a + bi) and its conjugate (a - bi) is:

(a + bi)*(a - bi) = a² + b²

Then in this case we can rewrite:

-x² - 6² = -(x² + 6²) = - (x + 6i)*(x - 6i)

                             = (-x - 6i)*(x - 6i)

The correct option is C.

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We have to use proportions to solve this question.

According to the given information, the perimeter of Figure A is to the sidelength of that figure as the perimeter of Figure B is to the sidelength of that figure:

The corresponding sidelength of Figure B is 30.

Let

the price of gummy bears per pound = x

the price of jelly beans per pound = y

price per pound of gummy bear = $3.2

price per pound of jelly beans = $7.5

Answer:

1/2 because half the cards are red and half the numbers are even

The rate at which the distance between the two cars increased four hours later is 30 mi/h.

First of all, we would determine the distances travelled by each of the cars. The distance travelled by the first car after four (4) hours is given by:

Distance, x = speed/time

Distance, x = 24/4

Distance, x = 6 miles.

For the second car, we have:

Distance, y = speed/time

Distance, y = 18/4

Distance, y = 4.5 miles.

After four (4) hours, the total distance travelled by the two (2) cars is given by this mathematical expression (Pythagorean theorem):

z² = x² + y²

Substituting the parameters into the mathematical expression, we have;

z² = 6² + 4.5²

z² = 36 + 20.25

z² = 56.25

z = 7.5 miles.

Next, we would differentiate both sides of the mathematical expression (Pythagorean theorem) with respect to time, we have:

2z(dz/dt) = 2x(dx/dt) + 2y(dy/dt)

Therefore, the rate of change of speed (dz/dt) between the two (2) cars is given by:

dz/dt = [x(dx/dt) + y(dy/dt)]/z

dz/dt = [6(24) + 4.5(18)]/7.5

dz/dt = [144 + 81]/7.5

dz/dt = 225/7.5

dz/dt = 30 mi/h.

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Answer:

30 mi / hr

Step-by-step explanation:

First find out how far the cars are apart after 4 hours

24 * 4 = 96 mi = y

18 * 4 = 72 mi = x

Now use the pythagorean theorem

   s^2 =  ( x^2 + y^2 )                shows s = 120 miles apart at 4 hours

Now s^2 = x^2 + y^2     Differentiate with respect to time ( d  / dt )

       2 s ds/dt    =  2x dx/ dt  + 2y dy / dt

             ds/dt    =   (x dx/dt + y dy/dt)/s

                           =  (72(18)  + 96(24)) / 120

               ds/dt   = 30 mi/hr

Given:

Exponential equation for the black bird is,

Required:

To find the starting point of bird and graph the given function.

Explanation:

(1)

The bird starting point is at x = 0,

(2)

The graph of the function is,

Final Answer

(1) 1.004

(2)

If angle "a" measures 42° the the other angles that will be congruent to angle "a" and also measure 42° will be angle d, angle e and angle h .

In the question ,

a figure is given ,

From the figure we can see that 2 parallel lines are cut by a transversal .

So ,

angle a = angle d   .......because vertically opposite angles .

angle a = angle e   ...because corresponding angles are equal in measure

also

angle e = angle h      .... because vertically opposite angles .

Therefore , If angle "a" measures 42° the the other angles that will be congruent to angle "a" and also measure 42° will be angle d, angle e and angle h , the correct option is (a) .

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The solution of the inequality is [tex]x \geq 289[/tex].

What is inequality?

Inequalities specify the connection between two non-equal numbers. Equal does not imply inequality. Typically, we use the "not equal sign ( [tex]\neq[/tex]) " to indicate that two values are not equal. But several inequalities are utilised to compare the numbers, whether it is less than or higher than.

The given inequality is,  [tex]\sqrt{x} \geq 17[/tex]

Taking square on both sides, we get

[tex]x\geq 289[/tex].

Therefore, the solution of the inequality is [tex]x \geq 289[/tex].

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The form of the equation of the circle is

Where (h, k) are the coordinates of the center

r is the radius

Since the endpoints of the diameter are (11, 2) and (-7, -4), then

The center of the circle is the midpoint of the diameter

The center of the circle is (2, -1), then

h = 2 and k = -1

Now we need to find the length of the radius, then

We will use the rule of the distance between the center (2, -1) and one of the endpoints of the diameter we will take (11, 2)

Now substitute them in the rule above