The radioactive element carbon-14 has a half-life of 5750 years. A scientist determined that the bones from a mastodon had lost 73.6% of their carbon-14. How old were the bones at the time they were discovered?
The bones were about how many years old?

The solution for the variable x is x = 8.4

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

The right triangle

Using the proportion of similar triagles, we have

x/7 = 10/x

This gives

x * x = 7 * 10

So, we have

x² = 70

Next, we have

x = √70

Evaluate

x = 8.4

Hence, the solution for the variable x is x = 8.4

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Answer:

C Exponential

Step-by-step explanation:

3x^2 -- its using an exponent

The shortest path from vertex A to vertex L in the given diagram is ABDGJLKFI, with a total length of 89.

We can find the length of the shortest path from vertex A to vertex L by calculating the sum of the shortest paths from A to C to F to I to L.

Using Dijkstra's algorithm, we can find the shortest path from A to C, then from C to F, then from F to I, and finally from I to L.

The shortest path from A to C is AB + BC = 3 + 16 = 19. The shortest path from C to F is CF = 25. The shortest path from F to I is FI = 21. The shortest path from I to L is IL = 24.

Therefore, the length of the shortest path from A to L is 19 + 25 + 21 + 24 = 89.

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Therefore , the solution of the given problem of expressions comes out to be there are roughly 7,000 grains of sand in 7 grammes of sand.

Instead of using random estimates, it is preferable to use shifting numbers that may also prove increasing, reducing, variable or blocking. They could only help one another by trading tools, information, or solutions to issues. The justifications, components, or quantitative comments for tactics like further disagreement, production, and blending may be included in the assertion of truth equation.

Here,

A kilogramme (1000 grammes) of sand is thought to contain around 1 million grains of sand. This indicates that 1 gramme of sand contains roughly 1000 grains of sand.

Using this calculation, we can determine how many sand grains there are in 7 grammes roughly as follows:

=> Number of sand grains in 7 grammes = (# of sand grains in 1 gramme) x (7 grams)

=> 1000 x 7 = the number of sand grains in 7 grammes.

=> 7,000 sand grains are included in 7 grammes.

In accordance with this calculation, there are roughly 7,000 grains of sand in 7 grammes of sand.

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The answer in exponent form is 147x^29.

A scientific notation is a method of expressing very large numbers in the form of exponent.

In the given question, we have;

(7x^9)^2*3x^11

applying the law of indices,

(7x^9)^2*3x^11 = (7^2)*x^(9*2)*3x^11

                        = 49x^18*3x^11

                        = (49*3)*x^(18+11)

                        = 147x^29

Thus, the expression can be simplified as: 147x^29.

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Mr. Moketsi could help students in their learning of properties of 2D shapes through hands-on activities and discovery, as suggested by the constructivist perspective.

Mr. Moketsi initiates the class by inquiring about their comprehension of 2D shapes and their properties, provoking conversation amongst students as they share their thoughts with their peers and their teacher.

He subsequently introduces an assortment of manipulatives, such as pattern blocks, geoboards, and tangrams to encourage experimentation and inquiry into these geometric figures, while posing thought-provoking questions throughout the lesson to help learners make connections between different shapes with a focus on discovering their unique attributes.

Mr. Moketsi encourages reflection, questioning how this newfound knowledge may be applied in reality, stimulating critical thinking abilities during classroom interaction with surfaces representing planes containing lengths and widths.

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Answer:

37

Step-by-step explanation:

We are given the system of equations:

y = 4x^2 + 3x - 5

y = 2x^2 + x + 7

We can set the two equations equal to each other to get:

4x^2 + 3x - 5 = 2x^2 + x + 7

Simplifying, we get:

2x^2 + 2x - 12 = 0

Dividing both sides by 2, we get:

x^2 + x - 6 = 0

Factoring the left-hand side, we get:

(x + 3)(x - 2) = 0

Therefore, the solutions are x = -3 and x = 2.

To find the corresponding y-coordinates, we can plug these values of x into either of the original equations. Using the first equation, we get:

y = 4(-3)^2 + 3(-3) - 5 = 16

and

y = 4(2)^2 + 3(2) - 5 = 21

Therefore, the sum of the y-coordinates of the solutions is:

16 + 21 = 37

Answer: 37.

In general, an antiderivative of a function f(x) is a function F(x) whose derivative is equal to f(x). The antiderivative of a polynomial function is obtained by adding one to the exponent of each term and dividing by the new exponent ⁵.

Answer:

1. f. System of Equations

2. e. No solutions

3. i. One solution

4. h. Like Terms

5. b. Substitution

6. c. Solution

7. a. Elimination

8. d. Infinitely Many Solutions

9. g. Variable

10. j. Coefficients

Step-by-step explanation:

Ur welcome :)

The remaining ribbon will be 14 feet.

Olga decorates blankets with ribbon she has 12 yards of ribbon

and, she uses 22 feet of the ribbon to decorates blankets

Now, we have to find the she decorates the blankets how many feet of ribbon will remain?

Firstly, Convert the yard into feet

We know that:

There are 3 feet in 1 yard

So, 36 feet in 12 yards

Now, The remaining ribbon will be the original amount less the amount used.

=> 36 - 12 = 14 feet

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Answer:

7

Step-by-step explanation:

We can use the fact that the corresponding sides of similar triangles are in proportion to each other.

Let's compare the corresponding sides of the two triangles:

Corresponding sides and ratio

Ratio

Since the triangles are similar, we know that these ratios are equal. That is,

6 / (x-3) = 12 / (x+1)

We can simplify this equation by cross-multiplying:

6(x+1) = 12(x-3)

Expanding and simplifying:

6x + 6 = 12x - 36

42 = 6x

x = 7

Therefore, x has a value of 7.

The p-value from the hypothesis is 0.0583 which is not significant at p<0.05

To determine the p-value of the hypothesis, we need to use the formula;

z = (x - a) / √(a(1 - a)/n)

x = 93 / 900 = 31/300 = 0.103

a = 12% = 12/100 = 0.12

n = 900

Substituting the values into the formula;

z = (0.103 - 0.12)/√(0.12(1 - 0.12)/900)

z = -0.017 / √0.0001173

z = -1.569

Using a standard normal distribution table;

p-value = 0.058324

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All four angles of the rhombus are 20, 160, 20, and 160 degrees.

Diagonals are the parts of a shape, in geometry, a diagonal is a line that join two vertices of a polygon or a solid shape , whose vertices are not on the same edge. In general, a diagonal is called  as a sloping line or the slant line, that connects to the vertices of a shape.

According to question a rhombus , all four sides are congruent and opposite angles are equal. Let's call the rhombus ABCD, where AB, BC, CD, and DA are the four sides, and AC and BD are the diagonals.

If one of the angles formed by diagonals and adjacent sides is 20 degrees, let's say that angle is ABD. We can draw a line from point A to point C, which will divide the rhombus into two congruent triangles, ABC and ACD.

Since triangle ABC is isosceles, with sides AB = BC, angles ABC and BCA are equal. Similarly, in triangle ACD, angles ACD and CDA are equal.

∴ angles ABD and CDA are adjacent angles formed by intersecting lines, they must add up to 180 degrees. Therefore, angle CDA is 160 degrees.

If triangles ABC and ACD are congruent, angle CAD is also 20 degrees.

Now we can use the fact that opposite angles in a rhombus are equal. Angles BCD and ABD are opposite angles, so angle BCD is also 20 degrees. Similarly, angles BAC and ADC are opposite angles, so angle BAC is also 160 degrees.

Therefore, all four angles of the rhombus are 20, 160, 20, and 160 degrees.

Complete question  If one of the angles formed by diagonals and adjacent sides. of Is 20, all four angles of the a rhombus are.

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In order to properly solve a system of linear equations utilizing matrices, you must abide by the below instructions:

First, create a matrix representation of the system of equations.  

Second, generate an augmented matrix by merging the coefficient matrix together with the constant matrix.

Third, apply elementary row operations in order to transform the augmented matrix into either row echelon or reduced row echelon form.

Fourth, use back substitution to accurately determine the variable values.

Ultimately, enter this solution into the original equation in order to obtain the primary variable value.

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Solving linear equations using matrices involves steps ranging from creating an augmented matrix, performing row operations, back substitution, and then interpreting results

To solve a system of linear equations using matrices, follow these step-by-step instructions:

Step 1: Write the system of linear equations in matrix form.
Represent the coefficients of the variables as a matrix (called the coefficient matrix), and the constants on the right side of the equations as another matrix (called the constant matrix).

Step 2: Create the augmented matrix.
Combine the coefficient matrix and the constant matrix into a single matrix by appending the constant matrix as an additional column. This combined matrix is called the augmented matrix.

Step 3: Perform row operations to achieve row-echelon form.
Use row operations (swapping rows, multiplying a row by a constant, or adding/subtracting rows) to manipulate the augmented matrix into row-echelon form. Row-echelon form has zeroes below the diagonal and non-zero elements on the diagonal.

Step 4: Perform back-substitution.
Starting from the last row of the row-echelon form matrix, solve for the variables using back-substitution. Substitute the values of the variables you find into the previous rows to determine the remaining variable values.

Step 5: Interpret the results.
Once you have solved all the variables, you have found the solution to the system of linear equations. If there are infinitely many solutions or no solutions, this will be indicated by the row-echelon form.

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You must measure 24 male students to estimate the mean height within one inch with 93% confidence.

Since we want to estimate the mean height within one inch, the margin of error (E) is 1 inch:

E = Z * (Standard deviation / √Sample size)

Now, we can rearrange the formula to solve for the Sample size (n):

1 = 1.81 * (2.7 / √n)

Solving for n:

n = (1.81 * 2.7 / 1)²

n ≈ (4.887)²

n ≈ 23.88

Since we cannot have a fraction of a person, we round up to the nearest whole number:

n ≈ 24

So, you must measure 24 male students to estimate the mean height within one inch with 93% confidence.

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Answer:90% probability

Step-by-step explanation:

To calculate the experimental probability of the cup landing on its side, we can use the formula:

experimental probability = number of times the cup landed on its side / total number of trials

For the first set of trials, the experimental probability is:

experimental probability = 19/20 = 0.95

For the second set of trials, the experimental probability is:

experimental probability = 81/90 = 0.9

So the experimental probability of the cup landing on its side for the first set of trials is higher than for the second set of trials. However, since the number of trials is relatively small, it is possible that this difference is due to chance and does not reflect a real difference in the probability of the cup landing on its side. To determine this with more confidence, we would need to conduct more trials and perform statistical tests to determine whether the difference is statistically significant.

The calculated length of XY in the right triangle XYZ is 6

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

The right triangle

The value of XY in the right triangle can be calculated using the following sine ratio

sin(30) = XY/XZ

So, we have

sin(30) = XY/12

Cross multiply the equation

So, we have

XY = 12 * sin(30)

Evaluate the products

XY = 6

Hence, the length of XY in the right triangle is 6

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The result of the logarithm function is equal to 9. (Correct choice: D)

The statement shows an incomplete logarithm, whose result must be found by understing relationships between trascendent functions. Logarithm functions and exponential functions are related in the following way:

aᵇ = n ↔ ㏒ₐ n = b

Where:

If we know that (base) a = 3 and (exponent) b = 2, then the value of the result (n) is now defined:

n = 9

3² = 9

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Answer:

A

Step-by-step explanation:

The wrestler's weight at the first match will be 152.1 pounds. The solution has been obtained by using the substitution method.

What is the substitution method?

The substitution method is a strategy used in algebra to solve simultaneous linear equations. In this process, as the name suggests, the value of one variable from one equation is swapped in the second equation.

We are given that the given situation can be modeled by the equation

y = −1.1x + 162.

Now, we need to find weight at the first match which is 9 weeks away.

So, we get that x = 9.

By substituting this value in the equation, we get

⇒ y = -1.1 (9) + 162

⇒ y = -9.9 + 162

⇒ y = 152.1

Hence, the wrestler's weight at the first match will be 152.1 pounds.

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1. The linear regression equation is y = -1.17x+ 10.69

2. The correlation coefficient is -0.982.

3. This means there is a strong negative correlation between the gallons and hours.

4. The number of gallons is decreasing as the hours are increasing.

Step 1: find slope.

Formula for slope m = (n × Σ(xy) - Σx × Σy) / (n × Σ(x²) - (Σx)²)

Σx = 1 + 2 + 3 + 4 + 5 = 15

Σy = 9.2 + 8.4 + 7.7 + 6.1 + 4.5 = 35.9

Σ(xy) = (1 × 9.2) + (2 × 8.4) + (3 × 7.7) + (4 × 6.1) + (5 × 4.5) = 96

Σ(x²) = 1 + 4 + 9 + 16 + 25 = 55

n = 5

Therefore ∴

m = (5 × 96 - 15 × 35.9) / (5 × 55 - 15²) = -1.17

y-intercept b = (Σy - m × Σx) / n

b = (35.9 - (-1.17) × 15) / 5 = (53.15) / 5 = 10.63

Therefore the linear regression equation is y = -1.17x + 10.63

correlation coefficient (r): (n × Σ(xy) - Σx × Σy) / sqrt((n × Σ(x²) - (Σx)²) × (n × Σ(y²) - (Σy)²))

r = (5 × 96 - 15 × 35.9) / sqrt((5 × 55 - 15²) × (5 × 271.95 - 35.9²)) = -0.982

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The factored form of 64d³ - 24d² is 8d²(8d³ - 3).

The greatest common factor (GCF), also known as the greatest common divisor (GCD), is the largest positive integer that divides two or more numbers without leaving a remainder. In other words, it is the largest number that divides evenly into two or more given numbers. For example, the GCF of 12 and 18 is 6, since 6 is the largest number that divides both 12 and 18 without leaving a remainder.

The greatest common factor (GCF) of the given terms is 8d². We can factor it out by dividing each term by 8d²:

64d³/8d² - 24d²/8d²

Simplifying the fractions:

8d³ - 3

Therefore, the factored form of 64d³ - 24d² is 8d²(8d³ - 3).

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The distance between the points (6,−10) and (6,−7) is 3 units

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

Points (6,−10) and (6,−7)

Because the points have the same x coordinate, the distance between the points (6,−10) and (6,−7) is the difference between the y coordinates

Using the above as a guide, we have the following:

Distance = -7 - -10

Evaluate

Distance = 3

Hence, the distance between the points (6,−10) and (6,−7) is 3 units

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The correct function that models the balance B of Lily’s account as a function of time in months is option A, which is B(t) = 325(1 − 0.005)ᵗ.

To see why, we can start with the initial balance of $325 and note that each month, the bank charges a monthly fee of 0.5%, which is equivalent to a monthly interest rate of 0.005. This means that after one month, the balance will be reduced by 0.5% or 0.005 times the original balance, giving:

B(1) = 325 - 0.005(325)

Similarly, after two months, the balance will be reduced by another 0.5% of the new balance, giving:

B(2) = (325 - 0.005(325)) - 0.005(325 - 0.005(325))

We can simplify this expression by factoring out the common factor of 325(1 - 0.005) after each term, giving:

B(2) = 325(1 - 0.005)²

Generalizing this pattern, we can see that after t months, the balance will be:

B(t) = 325(1 - 0.005)ᵗ

Therefore, option A is the correct function that models the balance of Lily’s account.

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a. To build the airport at the same distance from the center of each town, they should locate the intersection of the perpendicular bisectors of the segments connecting each pair of towns.

b. To build the airport equidistant from each road, locate the intersection of the angle bisectors at each town

The precise location of the airport will be the point where these perpendicular bisectors intersect. This point will be equidistant from the center of each town.

b. To build the airport equidistant from each road, locate the intersection of the angle bisectors at each town. This point is the precise airport location. Equidistant from all connecting roads.

They could find the intersection of the medians of the triangle formed by the three towns. The airport is located at the centroid of a triangle, equidistant from 3 towns at the intersection of medians.

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Answer:

a) 192Km

b) 50

Step-by-step explanation: