Native Americans once lived near Cartersville, Georgia, in an area that is now a state park. They constructed burial mounds that often contained artifacts, such as beads, feathers, and copper ear ornaments. One of the park's mounds is 63 feet in height. Be a Math Detective. If the top of the mound is rectangular in shape with a perimeter of 322 yards, what could be the side lengths of the rectangle?

Answer:

The coordinates of A is  [tex](-x_1,-y_1)[/tex].

Step-by-step explanation:

We are given that a triangle ABC is rotated 180 degrees about the origin .

We have to find the coordinates of A.

Let vertices of triangle ABC [tex]A(x_1,y_1),B(x_2,y_2) ,C(x_3,y_3)[/tex]

When we rotate the about 180 degrees then the coordinates changes like as

[tex](x,y)\rightarrow (-x,-y)[/tex]

When we rotate triangle ABC about 180 degrees then its vertices Ais ([tex]x_1,y_1)[/tex] change into [tex](-x_1,-y_1)[/tex]

Hence, the coordinates of A is  [tex](-x_1,-y_1)[/tex].

Final answer:

After rotating triangle ABC 180 degrees about the origin, the coordinates of point A, initially at the origin (0, 0), remain unchanged at (0, 0).

Explanation:

If triangle ABC is rotated 180 degrees about the origin, the coordinates of point A after the rotation can be determined by applying the rules of rotation in the Cartesian coordinate system. Since point A is at the origin, its initial coordinates are (0,0). A rotation of 180 degrees about the origin will essentially reflect a point over both the x-axis and y-axis, but the location of point A will remain unchanged because it is located at the center of rotation. Therefore, the rotated coordinates of point A will still be (0, 0).

A rational number is a number which is in the form [tex]\frac{p}{q} ,q \neq 0[/tex]

Now, the value of the number [tex]\sqrt 3 = 1.73[/tex]

Now let us evaluate each given rational number in decimal form.

[tex]\frac{14}{9} = 1.6\\ \\ \frac{15}{13} = 1.2\\ \\ \frac{17}{10} = 1.7\\ \\ \frac{6}{5} = 1.2[/tex]

Therefore, among these numbers, the best approximation is option c i.e. 17/10.

C is the correct option.

Answer:

C) [tex]\frac{17}{10}[/tex]

Step-by-step explanation:

[tex]1,732050808 ≈ \sqrt{3} \\ \\ 1,73 ≈ \sqrt{3}[/tex]

Therefore answer choice C) would be the best approximate answer to this:

* [tex]1\frac{7}{10} = 1,7[/tex]

I am joyous to assist you anytime.

Rational Root Theorem

Final answer:

The theorems for finding zeros of higher degree polynomials include the Fundamental Theorem of Algebra, which guarantees n roots for an nth degree polynomial, the quadratic formula for second-order polynomials, and Euler's Theorem for Homogeneous Functions. These tools and theorems assist us in identifying possible roots and understanding the behavior of polynomial functions.

Explanation:

Theorems for Finding Zeros of Higher Degree Polynomial Functions

There are several important theorems relevant to finding zeros or solutions of higher degree polynomial functions. One key theorem is the Fundamental Theorem of Algebra, which states that every nth-degree polynomial has exactly n complex roots, which may include repeated roots. Additionally, polynomials are continuous and differentiable, so finding points where the derivative is zero leads to identifying potential maxima and minima.

The quadratic formula is used to find the zeros of second-order polynomials, revealing up to two roots, which may be real or complex numbers. Another relevant theorem is Euler's Theorem for Homogeneous Functions, which pertains to homogeneous functions of a certain degree and their properties. In the case of polynomials, it can be applied to determine certain types of symmetries and relationships amongst the coefficients.

Regarding polynomials of odd degrees, these always have at least one real root. Furthermore, since real-world numbers are approximations, tiny changes in the coefficients of a polynomial can lead to distinct roots. It's also worth noting that for any nth degree polynomial, differentiation yields an n-1 degree polynomial, which guides us in understanding the number of maxima or minima the function can possess. This is illustrated by the derivative of a third-order polynomial being a second-order polynomial, which can have at most two real roots.

If one angle measures 32 degrees, the measure of the second angle can be found by subtracting 32 from 90. Therefore, the second angle is 58 degrees.

Complementary angles are two angles whose sum is 90 degrees. So, if one angle measures 32 degrees, we can find the measure of the second angle by subtracting 32 from 90:

Second angle = 90 - 32

Second angle = 58 degrees

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The given problem is an algebraic equation where the daughter's age is denoted as 'x' and the father's age as '10x'. An equation is formulated to solve for 'x' based on future ages. The solution to the problem is that the daughter is 5 years old, and the father is 50 years old.

The problem asked is a classic example of algebra problem solving. We are told that a father is 'ten times as old as his daughter', and in five years, 'he will be just five times as old as she will be'. Let's denote the daughter's current age as 'x'. Therefore, the father's current age will be '10x'.

In five years, the daughter will be 'x+5' and the father will be '10x+5'. At this future moment, the father is 'five times as old as the daughter', so we can create the equation: 10x + 5 = 5(x + 5). Solving this equation leads to 'x=5'. So, the daughter is 5 years old and the father is 50 years old.

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The helicopter has flown a distance of 500 feet as calculated using the Euclidean distance formula for two points on a grid.

To find the distance the helicopter has flown, we need to calculate the Euclidean distance between the two points (2,6) and (6,9) on the grid map. We can use the formula: Distance = √[(x₂ - x₁)² + (y₂ - y₁)²], where (x₁, y₁) and (x₂, y₂) are the coordinates of the two points. Substituting the given points into the formula, we have:

Distance = √[(6 - 2)² + (9 - 6)²] = √[(4)² + (3)²] = √[16 + 9] = √25 = 5 grid lines.

Given that each grid line is equivalent to 100 feet, the helicopter has traveled 5 grid lines * 100 feet/grid line = 500 feet.

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Answer:

B. Stress

Step-by-step explanation:

Did the usa prept test :)

To find the probability that both marbles drawn by Joseph are black, you need to consider the number of ways he can draw 2 black marbles out of the 4 black marbles in the bag, divided by the total number of ways he can draw 2 marbles from the 10 marbles in the bag without replacement. The probability is 2/15.

To find the probability that both marbles drawn by Joseph are black, we need to consider the number of ways he can draw 2 black marbles out of the 4 black marbles in the bag, divided by the total number of ways he can draw 2 marbles from the 10 marbles in the bag without replacement.

Let's calculate:

Therefore, the probability that both marbles are black is 6/45, which simplifies to 2/15.

That’s what it looks like. Im not sure how to solve it either though.