92199+20923+29290+83292+2819+99279+38471+378141
= 744414
Frequency polygon question: work out the upper and lower bonds
They are not the same as the upper and lower limits used in other statistical calculations.
what are statistical data?A accumulation of numerical data that has been methodically gathered, arranged, and examined is referred to as statistical data. Numerous phenomena, including economic patterns, demographic traits, health outcomes, and more, are described and measured using this data. Numerous techniques, such as surveys, experiments, observations, and administrative documents, can be used to gather statistical data. After being gathered, the data can be examined statistically to reveal patterns, connections, and trends that can be used to guide policy development and decision-making.
given
This is a histogram showing the frequency distribution of the weights of 30 students. Each bar represents a weight range, and the height of the bar represents the frequency of students in that weight range.
To find the upper and lower boundaries for each weight range, we can use the following formula:
Lower bound = midpoint of the previous range + 0.5
Upper bound = midpoint of the current range - 0.5
For example, the lower bound of the first range (60-64) is:
Lower bound = midpoint of previous range + 0.5
= 57
Upper bound = midpoint of current range - 0.5
= 61.5
We can apply the same formula to find the upper and lower bounds for the other ranges in the histogram. The results are:
60-64: lower bound = 57, upper bound = 61.5
65-69: lower bound = 62, upper bound = 66.5
70-74: lower bound = 67, upper bound = 71.5
75-79: lower bound = 72, upper bound = 76.5
80-84: lower bound = 77, upper bound = 81.5
85-89: lower bound = 82, upper bound = 86.5
90-94: lower bound = 87, upper bound = 91.5
95-99: lower bound = 92, upper bound = 96.5
They are not the same as the upper and lower limits used in other statistical calculations.
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The diameter of a circle is 38 centimeters. What is the circle's circumference?
Use 3.14 for л.
Answer: 119.32 centimeters
Step-by-step explanation:
Circumference = diameter x π
C = 38 (3.14)
C = 119.32
The circumference is 119.32 centimeters
Hope this helps!
Amy is attending a school orchestra concert.she sees her math teacher seated 10 meters ahead of her and her science teacher seated 24 meters to her right.How far apart are the two teachers?
Answer:
The two teachers are 26 meters apart. To calculate this, you can use the Pythagorean Theorem, which states that the distance between two points is equal to the square root of the sum of the squares of the differences between the coordinates of the two points. In this case, the coordinates of the math teacher are (10, 0) and the coordinates of the science teacher are (0, 24), so the distance between them is the square root of [(10-0)^2 + (0-24)^2] = √(100 + 576) = √676 = 26 meters.
We can use the Pythagorean theorem to solve this problem. Let's draw a diagram:
A (Amy)
/|
/ |
/ |10 meters
/ |
/____|
B C (science teacher)
We have a right triangle ABC, where AB = 10 meters and BC = 24 meters. We want to find the length of AC, which is the distance between the two teachers.
Using the Pythagorean theorem:
AC^2 = AB^2 + BC^2
AC^2 = 10^2 + 24^2
AC^2 = 676
AC = sqrt(676)
AC = 26 meters
Therefore, the two teachers are 26 meters apart.
Look at the following table then answer the questions below
a. Which of the functions in the table appears to be exponential?
b. What reasoning would you use to justify your answer?
c. Which function(s) would most likely model bacterial growth in a lab culture? Justify your reasoning.
d. Which values would most likely model a tub collecting water from a leaky faucet? Justify your reasoning.
The evaluation of the functions for data in the table and the graphs of the data, created using MS Excel, indicates that we get;
a. Equation 3 appears to be exponential
b. The reason is that as the value of x increases, the output values of the equation increases exponentially
c. The population of a bacteria is most likely modeled by Equation 3, which is an exponential function, because as the number of bacterium increases, the number of offspring they can produce in the next period increases, resulting in an exponential growth
d. Equation 4, which can be modeled using a linear equation, with a positive slope most likely models a tub collecting water from a leak faucet.
How can a function model an observed occurrence?A function can be used to model an observation by specifying the relationship between the input and output values of the observation.
a. The function in the table that appears to be exponential is Equation 3
b. An exponential function is a function that has an equation of the following forms;
f(x) = [tex]a\cdot b^{c\cdot x}[/tex]
f(x) = [tex]p\cdot e^{k\cdot x}[/tex]
The values in the table for Equation 3 indicates that the as the value of x increases, the values of f(x) increases several times more, which is an exponential increase, such that an equation that can model the data, obtained using MS Excel is; f(x) = 0.1982 × e^(0.3092·x)
Plugging in the values of x, we get;
The value of f(x) at x = 9 is; f(9) = 0.1982 × e^(0.3092 × 9) ≈ 3.2
The value of f(x) at x = 10 is; f(10) = 0.1982 × e^(0.3092 × 10) ≈ 4.36
The value of f(x) at x = 17 is; f(17) = 0.1982 × e^(0.3092 × 17) ≈ 38.0
Therefore, the function for Equation 3 appears to be exponential
Please find attached the graph for the Equation 3 data values, created with MS Excel
c. The growth of a population is best modelled by an exponential function, therefore, the function that most likely models a bacteria growth in a lab culture is Equation 3.
d. The tub collecting water from a leaking faucet collects water at the steady rate at which the faucets leaks, which can be modelled by a linear equation with a positive slope, because the amount of water in the tub increases steadily.
The graph and trendline obtained from values of the data for Equation 4, indicates that the line of best fit is a line with the equation;
y = 0.0626·x + 2.0016The square of the regression coefficient is; R² = 0.9993
The slope of the line of best fit of Equation 4 is positive, and therefore increasing at a steady rate and can therefore model collecting water from a leaky faucet using a tub
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points F(2,6), G(2,-1). H(x,y) form a triangle in the xy -coordinate plane. if the area of this triangle is 14 square units, then which of the following are possible coordinates for point H?
The perpendicular line connecting points F and G results in the formation of a triangle with a 14 square unit surface area.
What does coordinate mean?In mathematics, a coordinate is a set of numbers or symbols that describes an object's position or location in a geometric space. The most popular coordinate system is the Geographic coordinate system, which creates a grid on a surface or in three dimensions using a series of perpendicular lines. Each location on the plane or in space is then uniquely identified by an ordered pair or set of three numbers, respectively, that represent a point's distance from the origin along each of the coordinate system's axes.
Given that points F and G share the same x-coordinate, the line connecting them is perpendicular. A degenerate triangle with negative area will be formed by any point H that shares the same x-coordinate as F and G.
D = √((2-2)2 + (6-(-1),2) = √(49), which = 7.
distance(FH)/7 = 2/7.
distance(GH)/7 = 4/7
Distance(FH) / Distance(GH) = 1 / 2.
The distance between any two points H(x, y) and F(2,6) or G(2,-1) can be calculated using the distance formula as follows:
distance(FH) is = √((x-2) + (y-6)).
Distance (GH) is = √((x-2) + (y+1)).
By changing the aforementioned values, we obtain:
√((x-2)**2**y+1**2)/7) = 4/7
(x-2)² + (y+1)² = 16
(x-2)² + (y-6) (y-6)² = 1/4 ((x-2)² + (y+1)²)
By condensing and extending, we obtain:
3(x-2)² + 13(y-6)² = 196
5(x-2)² + 25(y+1)² = 784
7(x-2)² + (y-6)² = 1
A triangle with a 14 square unit surface area is created when the perpendicular line joining F and G.
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what happens as x approaches zero from the negative direction
Therefore , the solution of the given problem of function comes out to be the limit may not exist, if the function is not continuous at x = 0.
What does the term function mean?The mathematical discipline encompasses arithmetic, symbols, and their divisions, as well as histology, construction, and both real and imagined geographic places. A function describes the connections between various elements, which all have an associated outcome. A function is composed of an array of different parts that, when combined, result in particular outcomes for each input.
Here,
We must be aware of the precise function or expression under consideration in order to respond to this query.
The action as x moves towards zero from the negative direction can change depending on the function.
If a function is continuous at x = 0, then in general, if the function has a limit as x gets smaller from the downward direction, however this limit will be the same as the limit as x reaches zero from the positive direction.
The limit as x approaches zero from the downward direction may vary from the limit as x approaches zero from the better way, or the limit may not exist, if the function is not continuous at x = 0.
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Find each value or measure.
The answers are explained in the solution below.
What is circle?A circle is a round-shaped figure that has no corners or edges. In geometry, a circle can be defined as a closed, two-dimensional curved shape.
Given are some circles, we need to find the value of x, in each,
Using the properties of the circle, If two chords in a circle are congruent, then their intercepted arcs are congruent, [the main theorem]
1) RS = 59 and ST = 10x-31,
10x-31 = 59
10x = 100
x = 10
2) arc JK = arc ML
Therefore,
7x-39 = 87
7x = 126
x = 18
3) arc AB = arc DC
Therefore,
2(13x-21) = 360°-(arc AD+arc BC)
2(13x-21) = 244
13x-21 = 122
13x = 143
x = 11
4) LM = NP
Therefore,
41-2x = 7x+5
36 = 9x
x = 4
LM = 41-2(4)
= 41-8 = 33
5) arc UV = arc VW
Therefore,
8x-17 = 5x+52
3x = 69
x = 23
arc WV = 5(23)+52 = 167
6) We know, that the distance of two equal chords are same from the center of a circle,
Therefore,
HJ = JI
3x+20 = 15x-64
84 = 12x
x = 7
JI = 105-64
JI = 41
7) Using the converse of the theorem used in question 6, we have,
Chord BC = Chord CD
Again using the main theorem of the question,
arc BC = arc CD
9x-53 = 2x+45
7x = 98
x = 14
arc BAD = 360°-(arc BC + arc CD)
arc BAD = 214°
8) arc LM = arc NP
8x-56 = 5x+22
3x = 78
x = 26
Therefore, arc LM = arc NP = 152°
m arc LP = 360°-(arc LM + arc NP + arc MN)
m arc LP = 17°
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The usual price of a bag is $61.00. If the bag is on a 50% offer, how much will it cost?
Answer: $30.50
Step-by-step explanation:
To find a discount, the formula is, List price - (List price x (percentage / 100))
Find a basis for the orthogonal complement of the subspace of R4 spanned by the following vectors. v1 = (1, −1, 7, 5), v2 = (2, −1, 1, 6), v3 = (1, 0, −6, 1) The required basis can be written in the form {(x, y, 1, 0), (z, w, 0, 1)}. Enter the values of x, y, z, and w (in that order) into the answer box below, separated with commas.
The values of x, y, z, and w are 6, -13, -1, and 4, respectively. So, the answer is 6, -13, -1, 4.
To find a basis for the orthogonal complement of the subspace of R4 spanned by the given vectors, we need to find the null space of the matrix formed by the given vectors. The matrix is:
```
1 -1 7 5
2 -1 1 6
1 0 -6 1
```
We can use the reduced row echelon form to find the null space of this matrix. The reduced row echelon form of this matrix is:
```
1 0 -6 1
0 1 13 -4
0 0 0 0
```
The null space of this matrix is the set of all vectors (x, y, z, w) such that:
```
x - 6z + w = 0
y + 13z - 4w = 0
```
We can write the null space in parametric form as:
```
x = 6z - w
y = -13z + 4w
z = z
w = w
```
We can write the null space in the form {(x, y, 1, 0), (z, w, 0, 1)} by setting z = 1 and w = 0 in the first vector, and setting z = 0 and w = 1 in the second vector. This gives us:
```
x = 6(1) - 0 = 6
y = -13(1) + 4(0) = -13
z = 6(0) - 1 = -1
w = -13(0) + 4(1) = 4
```
Therefore, the basis for the orthogonal complement of the subspace of R4 spanned by the given vectors is {(6, -13, 1, 0), (-1, 4, 0, 1)}. The values of x, y, z, and w are 6, -13, -1, and 4, respectively. So, the answer is 6, -13, -1, 4.
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Find the y-intercept of line y=4/3*+2/3
Write your answer as an integer or as a simplified proper or improper fraction, not as an ordered pair.
Answer:
y = (4/3)x + 2/3
The y-intercept is the value of y when x is equal to zero. So we can substitute x = 0 into the equation and solve for y:
y = (4/3)(0) + 2/3 = 2/3
The y-intercept of the line y = (4/3)x + 2/3 is 2/3.
Step-by-step explanation:
Answer:
The y-intercept is (2/3)
Step-by-step explanation:
I will assume that the equation is missing the "x," and will rewrite is as:
y = (4/3)x + (2/3)
This is in standard form of y = mx + b, where m is the slope and b is the y-intercept.
The slope is (4/3) and the y-intercept is (2/3)
\[ \left(\frac{3 \sqrt[3]{x y} y^{-3}}{\sqrt[3]{y} x^{-\frac{4}{3}}}\right)^{-1} \] (d) \( (\sqrt{y}+2)(\sqrt{y}-2) \)
The correct option is (c) $768x^{\frac{1}{3}}$
The given expression is as follows; \[ \left(\frac{3 \sqrt[3]{x y} y^{-3}}{\sqrt[3]{y} x^{-\frac{4}{3}}}\right)^{-1} \]Firstly, simplify the numerator and the denominator of the expression.\[\frac{3\sqrt[3]{xy}y^{-3}}{\sqrt[3]{y}x^{-\frac{4}{3}}}=\frac{3}{\sqrt[3]{x}x^{-\frac{4}{3}}\sqrt[3]{y}y^{-3}}\]Next, simplify the denominator of the expression.\[\frac{3}{\sqrt[3]{x}x^{-\frac{4}{3}}\sqrt[3]{y}y^{-3}}=3x^{\frac{1}{3}}y^{\frac{10}{3}}\]Now, let us substitute $y=16$ into the expression of $3x^{\frac{1}{3}}y^{\frac{10}{3}}$.\[3x^{\frac{1}{3}}y^{\frac{10}{3}}=3x^{\frac{1}{3}}(16)^{\frac{10}{3}}=48x^{\frac{1}{3}}(2)^{\frac{10}{3}}=768x^{\frac{1}{3}}\]As we obtained the value of the expression by substituting $y=16$, therefore the correct option is (c) $768x^{\frac{1}{3}}$.
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The following table gives the frequency distribution of waiting
times (in minutes) for these customers.
The mean waiting time is:
The mean waiting time for these customers is 18.33 minutes.
The mean waiting time can be calculated by multiplying each waiting time by its frequency, summing up these products, and then dividing by the total number of customers.
In mathematical terms, the mean waiting time is:
Mean = (x1 * f1 + x2 * f2 + ... + xn * fn) / (f1 + f2 + ... + fn)
Where x1, x2, ..., xn are the waiting times, and f1, f2, ..., fn are the corresponding frequencies.
Using the data from the table, we can plug in the values and calculate the mean waiting time:
Mean = (5 * 2 + 10 * 4 + 15 * 6 + 20 * 8 + 25 * 10) / (2 + 4 + 6 + 8 + 10)
Mean = (10 + 40 + 90 + 160 + 250) / 30
Mean = 550 / 30
Mean = 18.33 minutes
Therefore, the mean waiting time for these customers is 18.33 minutes.
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Determine the value of the annuity for the indicated monthly deposit amount, the number of deposits, and the interest rate. You will need to determine the value for r to solve this problem. When finding r round to the nearest hundredth. Deposit amount: $100; total deposits: 120; interest rate: 10%, compounded semi-annually
The value οf the annuity after 120 depοsits is $1,059,780.64.
What is cοmpοund interest?Cοmpοund interest is interest that is earned nοt οnly οn the principal amοunt but alsο οn any interest earned οn that principal amοunt οver time.
Tο determine the value οf the annuity, we can use the fοrmula fοr the future value οf an annuity:
[tex]A = P \times \dfrac{(1 + r/n)^{(nt)} - 1)}{(r/n)}[/tex]
where A is the future value οf the annuity, P is the mοnthly depοsit amοunt, r is the annual interest rate, n is the number οf times the interest is cοmpοunded per year, and t is the tοtal number οf depοsits.
In this case, P = $100, n = 2 (since interest is cοmpοunded semi-annually), t = 120, and r is the interest rate per year, sο we need tο find r. Tο dο this, we can use the fοrmula:
[tex]A = P\times (1 + r/n)^{(nt)[/tex]
where A is the future value of the annuity after t depοsits.
Substituting the given vaIues, we have:
[tex]\$100 \times (1 + r/2)^{(2\cdot120)[/tex] = A
SimpIifying, we get:
[tex](1 + r/2)^{240} = A/100[/tex]
Taking the natural lοgarithm of both sides, we get:
[tex]ln(1 + r/2)^{240} = ln(A/100)[/tex]
240*ln(1 + r/2) = ln(A/100)
ln(1 + r/2) = ln(A/100)/240
Solving for r, we get:
[tex]$r = 2 \times \frac{e^{(ln(A/100)}}{240) - 1}[/tex]
Substituting the given vaIues, we get:
r = 0.4902 or 49.02%
Now we can plug in all the values into the original fοrmula for the future value οf the annuity:
[tex]A = $100\times \dfrac{(1 + 0.4902/2)^{(2\cdot 120) - 1)}}{0.4902/2}[/tex]
SimpIifying, we get:
A = $1,059,780.64
Therefοre, the value οf the annuity after 120 depοsits is $1,059,780.64.
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A regular octagon is shown below. Suppose that the octagon is rotated clockwise about its center so that the vertex at T is moved to V. How many degrees does
the octagon rotate?
The octagon has rotated 45° when the vertex at T is moved to V. This is because the angle of rotation for a regular octagon is 45° when the vertex at T is moved to V.
What is an octagon?An octagon is a two-dimensional shape with eight sides and eight angles. It is a polygon which means it is a closed, two-dimensional shape with straight sides. Octagons are used in architecture and design and are commonly seen in stop signs and floor tiles. The angles of an octagon are all equal and each side is the same length.
The regular octagon has eight equal sides and angles. The angles of an octagon are all of the same size, which is 135°. If the octagon is rotated clockwise about its center, then the vertex at T is moved to V. This means that the octagon will have rotated 45° in order to move T to V.
To calculate the angle of rotation, we can use the formula: angle of rotation = (360°/number of sides). Therefore, the angle of rotation for this octagon will be (360°/8) = 45°.
To confirm this, we can use a protractor to measure the angle between the two lines. The angle between the two lines is 45°. This confirms that the octagon has rotated by 45° as the vertex at T has been moved to V.
Therefore, the octagon has rotated 45° when the vertex at T is moved to V. This is because the angle of rotation for a regular octagon is 45° when the vertex at T is moved to V. This can be confirmed by using a protractor to measure the angle between the two lines.
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1. Use the Wronskian to determine whether the following sets of functions are linealy inde- pendent (a) {cos(x), 3 cos(x) + sin(2x), sin(x)} (b) {e", ce", cº}
Yes, we can use the Wronskian to determine whether the following sets of functions are linearly independent. Let's take a look at each set:
Set (a): {cos(x), 3 cos(x) + sin(2x), sin(x)}
To determine if this set of functions is linearly independent, we calculate the Wronskian of the functions. The Wronskian of this set of functions is:
W(cos(x), 3 cos(x) + sin(2x), sin(x)) = -2 cos(2x) - 3 sin(x)
Since the Wronskian is not equal to zero, this set of functions is linearly independent.
Set (b): {e', ce', cº}
To determine if this set of functions is linearly independent, we calculate the Wronskian of the functions. The Wronskian of this set of functions is:
W(e', ce', cº) = e
Since the Wronskian is not equal to zero, this set of functions is also linearly independent.
In conclusion, both sets of functions are linearly independent.
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Solving a two-step linear inequality: Solve the inequality for x. -3>(x)/(4)-1 Simplify your answer as much as possible.
To solve the inequality -3 > (x)/(4) - 1, we need to isolate the variable x on one side of the inequality. We can do this by following these steps:
Step 1: Add 1 to both sides of the inequality to eliminate the constant term on the right side.
-3 + 1 > (x)/(4) - 1 + 1
-2 > (x)/(4)
Step 2: Multiply both sides of the inequality by 4 to eliminate the fraction on the right side.
-2 * 4 > (x)/(4) * 4
-8 > x
Step 3: Rewrite the inequality with the variable on the left side.
x < -8
Therefore, the solution to the inequality is x < -8.
Note: When multiplying or dividing an inequality by a negative number, the inequality symbol must be reversed. However, in this case, we multiplied by a positive number (4), so the inequality symbol remains the same.
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Define a scheme procedure that takes a set (a list of distinct elements) and generates a list of all subsets of the set. For example, (subsets '(a b c )) returns ((a b c) (b c) (a c) (c) (a b) (b) (a) ()). Then, manually trace your procedure with the provided example. Please study provided examples in the lecture notes to learn how you should manually trace our procedure
Scheme procedure to generate a list of all subsets of a set:
(define (subsets set)
(if (null? set)
'(())
(let ((rest (subsets (cdr set))))
(append rest (map (lambda (x) (cons (car set) x)) rest))))))
The subsets procedure takes a set as input and checks if the set is empty. If the set is empty, it returns a list with an empty set as its only element. Otherwise, it calls the subsets procedure recursively on the rest of the set (without the first element) and stores the result in a variable called rest.
It then appends the rest list to the result of mapping a lambda function over the rest list. The lambda function takes an element x from the rest list and conses the first element of the original set to it to create a new subset. This results in a list of all subsets of the original set.
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11x-10+80=180 solve for x
Answer:
To solve for x, we need to isolate it on one side of the equation by performing inverse operations.
11x - 10 + 80 = 180
First, we combine the constant terms on the left side of the equation:
11x + 70 = 180
Next, we isolate the variable term by subtracting 70 from both sides:
11x = 110
Finally, we solve for x by dividing both sides by 11:
x = 10
Therefore, the solution to the equation 11x - 10 + 80 = 180 is x = 10.
calculate the height of n of the flag pole
give your answer in meters to 1 dp
Using a trigonometric relation we will see that the height is 8.62m
How to get the height of the flag pole?On the diagram we can see a right triangle, where we know one of the angles and the adjacent cathetus and we want to find the opposite cathetus.
Then we can use the trigonometric relation:
tan(a) = (opposite cathetus)/(adjacent cathetus)
Replacing the values that we know we will get:
tan(72°) = n/2.8m
2.8m*tan(72°) =n = 8.62m
That is the height.
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without evaluating the expression, why is 3^4 -1 even
Without evaluating the expression, the sum of given expression is 80 is this why its even number.
What are expressions?Using letters or alphabets to represent numbers without giving their exact values is the idea behind algebraic expressions. The fundamentals of algebra taught us how to express an unknown value using letters like x, y, and z. These letters are referred to here as variables.
In an algebraic expression, both constants and variables can be used. Any amount that is added before a variable and then multiplied by it is referred to as a coefficient.
An algebraic expression in mathematics is one that contains variables, constants, and algebraic operations (addition, subtraction, etc.). Expressions are built upon terms.
Calculate the power: 81
Calculate the sum or difference: 81 - 1
Answer is 80
Thus, here we know that 80 is an even number.
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Given f (x)=7x-9, (a) Find f (x+h) and simplify. f(x+h)-f(x) (b) Find and simplify. h Part: 0/2 Part 1 of 2 (a) f(x+h)=
From equation f (x)=7x-9, f (x+h)= 7h and we can simplify it as 7.
For part (a), we can start by substituting x+h for x in the equation for f(x):
f(x+h) = 7(x+h) - 9
f(x+h) = 7x + 7h - 9
We can then simplify the expression to:
f(x+h) = 7x + 7h - 9
Now, to find f(x+h) - f(x), we can subtract f(x) from both sides of the equation:
f(x+h) - f(x) = 7x + 7h - 9 - (7x - 9)
f(x+h) - f(x) = 7h
Therefore, for part (a) we can conclude that f(x+h) - f(x) = 7h.
Now, we can simplify the difference quotient using f(x+h) and f(x):
f(x+h) - f(x) = [7(x+h) - 9] - [7x - 9]
= 7h
f(x+h) - f(x) / h = 7h / h = 7
We have already simplified the difference quotient to 7.
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PLEASE ANSWER THE 2 QUESTION USE ANY STRATEGIE
According to the above, the statement that is true about the expression can be inferred: Since the factor of 5/6 is less than 1, the product of 8 * 5/6 will be less than 8. Additionally, the measure of the giant mouse would be 0.76 meters.
How to find the correct statement about fractions?To find the correct statement about the fractions we must analyze each of the statements and check it according to the multiplication information. According to the above, we can establish that the factor of 5/6 is less than one because the division of these two numbers results in a number less than 1.
5 / 6 = 0.83So each number that we multiply by this fraction will give a result less than the same number. In this case if we multiply 8 by this fraction it would give us 6.66
8 * 0.83 = 6.66On the other hand, if we want to know the height of the giant mouse, we must establish what the size of the Chihuahua is. So we develop the fraction 2 1/3 = 2.33
To find the height of the giant mouse we must divide the height of the chihuahua by 3.
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Triangle ABC is enlarged with a scale factor of -2 and the origin of the centre to give triangle ABC . WORK OUT THE COORDINATES OF A and B
The coordinates of the triangle A'B'C' are A'(-2, -6), B'(-14, -2) and C'(-2, -2).
What is a dilation?Dilation is the process of resizing or transforming an object. It is a transformation that makes the objects smaller or larger with the help of the given scale factor. The new figure obtained after dilation is called the image and the original image is called the pre-image.
From the given graph, triangle ABC have A(1, 3), B(7, 1) and (1, 1).
Triangle ABC is enlarged with a scale factor of -2 and the origin of the Centre to give triangle A'B'C'.
Now, A(1, 3) → -2(1, 3)→A'(-2, -6)
B(7, 1) → -2(7, 1) → B'(-14, -2)
C(1, 1) → -2(1, 1) → C'(-2, -2)
Therefore, the coordinates of the triangle A'B'C' are A'(-2, -6), B'(-14, -2) and C'(-2, -2).
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Evaluate. Write your answer as a fraction or whole number without exponents. 6^-2
[tex]\frac{1}{36}[/tex]
Step-by-step explanation:Negative exponents can be manipulated and solved through exponent properties.
Exponent Properties
There are numerous different exponent properties that allow us to simplify expressions. However, for this question, the important property is known as the negative exponent property. This states that [tex]\displaystyle a^{-b}=\frac{1}{a^b}[/tex]. We can apply this same idea to the question we were given.
Solving
Through the negative exponent property, we know that we can make the exponent positive by taking the reciprocal.
[tex]6^{-2}=\frac{1}{6^2}[/tex]Remember that 6^2 = 36, then solve.
[tex]6^{-2}=\frac{1}{36}[/tex]So, the final answer is 1/36.
Write an equation for the description. A number x increase by 16 is 124
The equation for the description for given number is x + 16 = 124
"A number x increases by 16" means that we start with a certain number, which we don't know yet, and then add 16 to it. So, we can represent this as:
x + 16
The next part of the description tells us that the result of this addition is 124. So, we can set up an equation by equating this expression to 124, like this:
x + 16 = 124
This is the equation that represents the given description. To find the value of x, we can solve for it by subtracting 16 from both sides of the equation:
x = 124 - 16
x = 108
So, the value of x that satisfies the given description is 108.
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3. In montht=0, a small group of rabbits escapes from a ship onto an island where there are no rabbits. The island rabbit population,P(t), in monthtis given byP(t)=1+24(0.85)′1000a) How many rabbits escaped from the ship? Explain how you arrived at your answer. b) EvaluateP(6)andP(24)and explain their meaning in the context of the problem. Use complete sentences. c) Use your calculator to graphP(t)for0≤t≤80. Obtain a printout of this graph, label each axis with both a variable and a word label, and label the scale on each axis. d) Describe the shape of the graph in mathematical words (increasing, decreasing, local maximum, local minimum). e) Use the graph to describe the manner in which the rabbit population has changed since the rabbits escaped. f) Does the graph suggest the growth in population you would expect among rabbits on an island. Explain g) Use the graph to estimate how long it will take before there are 500 rabbits. Show this on the graph vou drew. Write the conclusion in sentence.
a) The number of rabbits that escaped from the ship is 24.
b) P(6) = 1 + 24 (0.85)^6 = 91.7, and P(24) = 1 + 24 (0.85)^24 = 436.8.
c) The graph of P(t) for 0 ≤ t ≤ 80 is an exponential curve that increases steadily over time.
d) The graph of P(t) is increasing.
e) The graph shows that the population of rabbits has increased exponentially since they escaped from the ship.
f) Yes, the graph does suggest the growth in population that would be expected among rabbits on an island. This is due to the steadily increasing nature of the graph.
g) The graph suggests that it will take approximately 65 months for the rabbit population to reach 500. This can be seen on the graph, as the y-axis value reaches 500 around the 65th month.
a)This can be calculated by taking the initial population, 1, and multiplying it by the given rate of growth, 0.85, and raising that to the 1000th power.
b) The meaning of these numbers in the context of the problem is that P(6) represents the total rabbit population after 6 months, and P(24) represents the total rabbit population after 24 months.
c)It is labeled with the variable t on the x-axis and the word "Population" on the y-axis, with each axis scaled appropriately.
f) Yes, the graph does suggest the growth in population that would be expected among rabbits on an island. This is due to the steadily increasing nature of the graph.
g)The graph suggests that it will take 65 months for the rabbit population to reach 500. This can be seen on the graph, as the y-axis value reaches 500 around the 65th month.
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A hot air balloon is 520m from the ground. A building is 450m tall. If the angle of elevation from the top of the building to the hot air balloon is 10, find the horizontal distance from the balloon to the building in meters
Answer: The horizontal distance from the hot air balloon to the building is approximately 2533.39 meters.
Step-by-step explanation:
We can use trigonometry to solve this problem. Let's draw a diagram to visualize the situation!!
=== begin diagram ===
B (balloon)
/|
/ |
/ | 520m
/ |
/θ |
/ |
/______|___ C (ground)
A D
=== end diagram ===
In the diagram, we have a hot air balloon at point B that is 520m from the ground at point C. We also have a building at point D that is 450m tall. The angle of elevation from point D to point B is 10 degrees (angle θ).
We want to find the horizontal distance between point B and point D (distance AB in the diagram).
To do this, we can use the tangent function:
tan(θ) = opposite/adjacent
In this case, the opposite side is the height of the building (450m) and the adjacent side is the horizontal distance we want to find (AB). We can rearrange the formula to solve for AB:
AB = opposite/tan(θ)
AB = 450m / tan(10°)
AB ≈ 2533.39m
Therefore, the horizontal distance from the hot air balloon to the building is approximately 2533.39 meters.
How long, to the nearest year, will it take an investment to
double if it is continuously compounded at 18% per year?
_______ yr
It will take 4 years for an investment to double if it is continuously compounded at 18% per year.
The time it takes for an investment to double if it is continuously compounded at 18% per year can be calculated using the formula:
T = ln(2)/r
Where T is the time in years, ln(2) is the natural logarithm of 2, and r is the annual interest rate.
Substituting the given values into the formula, we get:
T = ln(2)/0.18
T ≈ 3.85 years
Therefore, to the nearest year, it will take approximately 4 years for the investment to double if it is continuously compounded at 18% per year.
A neperian logarithm or also known as a natural logarithm, refers to that logarithm that has e as its base, which is worth: 2.718281828.
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Find all rational zeros, if any, of the following polynomial. P(x)=x^(4)+2x^(3)-7x-14
The zeros of P(x)=x^(4)+2x^(3)-7x-14 are 1, -2, and 2.
To find the rational zeros, we need to use the rational zero theorem. This theorem states that any rational zeros of a polynomial must be a factor of the constant term (in this case, -14) divided by a factor of the leading coefficient (in this case, 1).
So, the possible rational zeros of this polynomial are ±1, ±2, ±7, and ±14.
To confirm if these are indeed the zeros of the polynomial, we can plug each of these numbers into the polynomial and determine if the result is 0.
For example, when x=7, P(7) = 7^(4)+2(7^(3))-7(7)-14 = 2401+882-49-14 = 1720. Since the result is not 0, 7 is not a zero of the polynomial.
So when x=2,
P(2) = 2^(4)+2(2^(3))-7(2)-14
= 16+16-14-14 = 0.
Therefore, 2 is a zero of the polynomial.
By repeating this process for all possible rational zeros, we can determine that the zeros of this polynomial are 1, -2, and 2.
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£300 in the ratio of 10:20
Answer:
To divide £300 in the ratio of 10:20, we need to first add the two parts of the ratio to find the total number of parts:
10 + 20 = 30
This means that the ratio can be expressed as 10/30 and 20/30.
Next, we can use these ratios to find the amount of money for each part of the ratio:
10/30 x £300 = £100
20/30 x £300 = £200
Therefore, the amount of money for the parts of the ratio 10:20 would be £100 and £200 respectively.