Rewrite the following quadratic function in vertex form. Then, determine the axis of symmetry. y=5x^2+15x-2

Using continuity concepts, it is found that K = 320 makes the function continuous.

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A function f(x) is said to be continuous at x = a if:

[tex]\lim_{x \rightarrow a^{-}} = \lim_{x \rightarrow a^{+}} = f(a)[/tex]

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The piece-wise function given is:

[tex]f(x) = \frac{5x^4 - 20x^3}{x-4}, x \neq 4[/tex]

[tex]f(x) = K, x = 4[/tex]

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The definition for x different of 4 can be simplified, as follows:

[tex]f(x) = \frac{5x^4 - 20x^3}{x-4} = \frac{5x^3(x - 4)}{x - 4} = 5x^3[/tex]

Thus, the simplified definition is:

[tex]f(x) = 5x^3, x \neq 4[/tex]

[tex]f(x) = K, x = 4[/tex]

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For the lateral limits, x is close to but not exactly 4, so the first definition is used.

[tex]\lim_{x \rightarrow 4^-} = \lim_{x \rightarrow 4} 5x^3 = 5(4)^3 = 5 \times 64 = 320[/tex]

[tex]\lim_{x \rightarrow 4^+} = \lim_{x \rightarrow 4} 5x^3 = 5(4)^3 = 5 \times 64 = 320[/tex]

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For continuity, it is needed that:

[tex]f(4) = 320[/tex]

Thus:

[tex]K = 320[/tex] makes the function continuous.

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1. Another term for adding money to an account is - contribute

One should contribute to his account. This phrase means, one should add money to his account. Contribution is addition of money in account.

2. Which situation allows you to have the most saved ?

A) having a set amount set aside for savings each time you are paid- this can be the most possible answer. Technically a person should save 20% to 25% of his income and this amount increases with income. So, each time when you are paid, set aside the savings amount.

Answer:

Step-by-step explanation:

What is the measure of the central angle?

Answer : 140°

What ratio represents the measure of the central angle compared to the measure of the entire circle?

Answer7/18

If s = , what is the length of minor arc AB?

Answer : 14π

Final Answer:

The answers are:
1. B. 5.76 x 10¹⁴ miles
2. D. $15,494.70

Explanation:

Let's solve each of these questions step by step.
Question 1:
First, we'll calculate the distance to the star in miles, given its distance in light years.
One light year is approximately 5.88 x 10¹² miles.
The star is 9.8 x 10¹ light years from Earth.

To find the distance in miles, we multiply the distance in light years by the length of one light year in miles:
Distance in miles = (5.88 x 10¹² miles/light year) * (9.8 x 10¹ light years)

To multiply these numbers in scientific notation, we multiply the coefficients (5.88 and 9.8) and then add the exponents of 10 (12 and 1).
Coefficient multiplication: 5.88 * 9.8
Using a calculator or doing the multiplication by hand, we find that 5.88 * 9.8 is approximately 57.624.
Exponent addition: 12 + 1 = 13

So the distance in miles is approximately 57.624 x 10¹³ miles.

However, in scientific notation, the coefficient should be between 1 and 10. Therefore, we need to adjust the coefficient from 57.624 to 5.7624 and increase the exponent by 1 to keep the same value.
Distance in miles = 5.7624 x 10¹⁴ miles

To make it approximately and match the answer choices, we can round the coefficient to 5.76.
Distance in miles ≈ 5.76 x 10¹⁴ miles

The closest answer to our computed value is:
B. 5.76 x 10¹⁴ miles

Question 2:
Next, let's use the compound interest formula to find the balance in the account after 33 years.
The compound interest formula is:
[tex]A = P (1 + r/n)^{(nt)[/tex]
where:
A is the amount of money accumulated after n years, including interest.
P is the principal amount (the initial amount of money).
r is the annual interest rate (decimal).
n is the number of times that interest is compounded per year.
t is the time the money is invested for, in years.

We're given:
P = $1,600.00
r = 7% annual interest = 0.07 (as a decimal)
n = 2 (since interest is compounded semiannually)
t = 33 years

Plugging these values into the formula gives us:
[tex]A = 1600.00 * (1 + 0.07/2)^{(2*33)[/tex]
Let's break it down:

Interest rate per compounding period: 0.07/2 = 0.035
Number of compounding periods: 2*33 = 66

Now substitute these values:
A = 1600 * (1 + 0.035)⁶⁶

We calculate (1 + 0.035)^66 using a calculator:
(1 + 0.035)⁶⁶ ≈ (1.035)⁶⁶ ≈ 9.682

So the final balance is:
A ≈ 1600 * 9.682 ≈ 15491.2

The closest answer to our computed value is:
D. $15,494.70

Hence, the answers are:
1. B. 5.76 x 10¹⁴ miles
2. D. $15,494.70

Answer:

The real answer is 846 ^-^ I got it right

The volume of the rectangular prism is 105.57

A rectangular prism is a three-dimensional shape that has two at the top and bottom and four are lateral faces.

The volume of a rectangular prism=Length X Width X Height

Where l, h and w represent the length, height and width respectively.

Given that;

Length of the prism is 2 times the width L = 2w

the width is 4 times the height  w = 4h

Now let’s plug these values;

the surface area equation, it becomes;

A = 2(2w(w) + w(w/4) + w/4(2w))

A = 2(2w^2 + w^2/4 + w^2/2)

A = 1/2(7w^2)

792 x2 = 7w^2

w = 15.04

The volume of a rectangular prism=Length X Width X Height

V = 2w x w x w/4

V x 2 = w³

V = 15.04³/2

V = 105.57

Thus, the volume is 105.57.

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Answer-

[tex]\boxed{\boxed{L=\dfrac{g}{4\pi^2 f^2}}}[/tex]

Solution-

The equation for time period of a simple pendulum is given by,

[tex]T=2\pi \sqrt{\dfrac{L}{g}}[/tex]

Where,

T = Time period,

L = Length of the rod,

g = Acceleration due to gravity.

Frequency (f) of the pendulum is the reciprocal of its period, i.e

[tex]f=\dfrac{1}{T}\ \Rightarrow T=\dfrac{1}{f}[/tex]

Putting the values,

[tex]\Rightarrow \dfrac{1}{f}=2\pi \sqrt{\dfrac{L}{g}}[/tex]

[tex]\Rightarrow (\dfrac{1}{f})^2=(2\pi \sqrt{\dfrac{L}{g}})^2[/tex]

[tex]\Rightarrow \dfrac{1}{f^2}=4\pi^2 \dfrac{L}{g}[/tex]

[tex]\Rightarrow L=\dfrac{g}{4\pi^2 f^2}[/tex]

Solving Q1: Given BE = ED and AE = EC.

Considering triangles ΔAED and ΔCEB;

1. AE = EC (given)

2. ∠AED = ∠CEB (vertical angles)

3. ED = EB (given)

ΔAED ≡ ΔCEB (Side-Angle-Side congruency of triangles)

∠DAE = ∠BCE and AD = BC (using CPCTC)

⇒ AD║BC (Converse of Alternate Interior angles theorem)

Similarly consider triangles ΔABE and ΔCDE;

1. AE = CE (given)

2. ∠BEA = ∠DEC (vertical angles)

3. BE = DE (given)

ΔABE ≡ ΔCDE (Side-Angle-Side congruency of triangles)

∠ABE = ∠CDE and AB = CD (using CPCTC)

⇒ AB║CD (Converse of Alternate Interior angles theorem)

Since we have AB║CD, AB = CD and AD║BC, AD = BC.

Therefore, quadrilateral ABCD is a parallelogram.

****************************************************************************************************

Solving Q2: The parallelogram has the angle measures shown in the diagram.

It is clearly visible that both the triangles are Isosceles triangles, so opposite sides in each triangle are equal.

Consider two triangles given in the problem, we have two sets of congruent angles and one included side is common in both triangles.

Using Angle-Side-Angle congruence of triangles, both the triangles would be congruent too.

Using CPCTC, we can say opposite sides of quadrilateral would be congruent.

Therefore, all sides in given quadrilateral are equal.

Hence, Given quadrilateral is a Rhombus.

We know that a two-way frequency table is constructed when we have to deal with two variables ( which is also known as a bivariate data)

Here we have two variables as wore a tie or not and the second variable is: Received compliment or not.

Hence, on the basis of the information provided to us in the question we see that the two-way frequency table is obtained as follows:

                                               Wore tie          Do not wore tie     Total

Receive compliment                 25                         5                       30

Do not receive compliment       9                         11                       20

        Total                                 34                         16                       50

The considered data given about the frequencies of events can be used to fill the two-way frequency table. The values will be 25 and 5 in first row, and 9 and 11 in second row.

Suppose two dimensions are there, viz X and Y. Some values of X are there as [tex]X_1, X_2, ... , X_n[/tex] and some values of Y are there as [tex]Y_1, Y_2, ... , Y_n[/tex]

List them in title of the rows and left to the columns. There will be [tex]n \times k[/tex] table of values will be formed(excluding titles and totals), such that:

Value(ith row, jth column) = Frequency for intersection of  [tex]X_i[/tex]  and [tex]Y_j[/tex]  (assuming X values are going in rows, and Y values are listed in columns).

Then totals for rows, columns, and whole table are written on bottom and right margin of the final table.

For n = 2, and k = 2, the table would look like:

[tex]\begin{array}{cccc}&Y_1&Y_2&\rm Total\\X_1&n(X_1 \cap Y_1)&n(X_1\cap Y_2)&n(X_1)\\X_2&n(X_2 \cap Y_1)&n(X_2 \cap Y_2)&n(X_2)\\\rm Total & n(Y_1) & n(Y_2) & S \end{array}[/tex]

where S denotes total of totals, also called total frequency.

n is showing the frequency of the bracketed quantity, and intersection sign in between is showing occurrence of both the categories together.

We're given that:

If we take:

A = event of Lea wearing a tie on a day

A' = event of Lea not wearing a tie on a day

B = event of Lea getting a compliment on a day

B' = event of Lea not getting a compliment on a day

then, we're given that:

[tex]n(A' \cap B') = 11\\n(B) = 30\\n(A') = 16[/tex]

Also, total number of days = S = 50

On each of those 50 days, she either wore a tie or not.

If she didn't put on a tie on 16 out of 50 days, then she wore tie for 50-16=34 days.

Thus, we get n(A) = 34

Similarly, we get n(B') = 50 - n(B) = 50 - 30 = 20

[tex]\begin{array}{cccc}&A&A'&\rm Total\\B&n(B\cap A)&n(B\cap A')&n(B)=30\\B'&n(B' \cap A)&n(B' \cap A')=11&n(B')=20\\\rm Total & n(A)=34 & n(A')=16 & S=50 \end{array}[/tex]

Using the total of the third row, we get:

[tex]n(B' \cap A) + 11 = 20\\n(B' \cap A) = 9[/tex]

Similarly, now using second column's total, we get:

[tex]n(B \cap A) + n(B' \cap A) = 34\\n(B \cap A) + 9 = 34\\n(B \cap A) = 25[/tex]

Now using second row's total, we get:

[tex]25 + n(B \cap A') = 30\\n(B\cap A') = 5[/tex]

Thus, we get:

[tex]\begin{array}{cccc}&A&A'&\rm Total\\B&n(B\cap A)=25&n(B\cap A')=5&n(B)=30\\B'&n(B' \cap A)=9&n(B' \cap A')=11&n(B')=20\\\rm Total & n(A)=34 & n(A')=16 & S=50 \end{array}[/tex]

Thus, finally, we can say that:

                                                  Wore a tie,               Didn't wore a tie

Got a compliment                         25                                    5

Didn't got a compliment               9                                      11

Thus, the values will be 25 and 5 in first row, and 9 and 11 in second row.

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The correct equation for the relationship between the measures of the segments, angles, and arcs is [tex]PR^2[/tex] = PS * PT. The last option is the right choice.

Power of a Tangent: We know that the power of a point from an external point P to a circle is equal to the product of the lengths of the two tangents drawn from P to the circle (PQ * PR = PS * PT).

Identify the Tangents and Secant: In the given diagram, PR is the secant intersecting the circle at points T and S, and PQ and PS are the tangents drawn from P to points Q and S, respectively.

Apply the Power of a Tangent Theorem: Since PS and PT are tangents from P, we can apply the power of a tangent theorem to get:

PS * PT = PQ * PR.

Substitute PQ with PR: Since PQ and PR are both tangents drawn from the same point P, they have the same length (PQ = PR). Substituting PQ with PR in the equation, we get:

PS * PT = PR * PR.

Simplify the Equation: Simplifying the equation PR * PR = PS * PT, we get the final equation:

[tex]PR^2[/tex] = PS * PT.

Therefore, the equation [tex]PR^2[/tex] = PS * PT correctly represents the relationship between the measures of the segments, angles, and arcs in the given diagram.

The last option is the right choice.

Answer:

150

Step-by-step explanation:

The measure of the side IJ is the longest one so the largest angle is ∠H. Then the correct option is B.

The polygonal shape of a triangle has a number of sides and three independent variables. Angles in the triangle add up to 180°.

Three survey markers are located on a map at points H, I, and J. A triangle is formed by connecting these markers by string so that HI = 150 feet, HJ = 245 feet, and IJ = 365 feet.

If the side is the largest one then its opposite angle will be the largest one.

The measure of the side IJ is the longest one so the largest angle is ∠H. Then the correct option is B.

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