2. The Data Above Should Follow A Pattern. Based On This Pattern, How Could You Balance The 5 Kg Block (2024)

Solution:

Given:

From the above graph, we have the location of the point P to be:

[tex]P(0,0)[/tex]

When the point translates horizontally by 12 units to P', we have

[tex]\begin{gathered} (0+12,\text{ 0\rparen} \\ \Rightarrow P^{\prime}(12,\text{ 0\rparen} \end{gathered}[/tex]

Thus, we have the transformation to be as shown below:

SOLUTION

From the question Yolanda rolled the cube 100 times. So if we sum the number of rolls, we get 100 which is the total number of outcomes

(a) The probability of rolling a 5 becomes outcome for 5, which is 19 divided by total number of outcomes, so the answer is

[tex]\begin{gathered} \frac{19}{100} \\ =0.19 \end{gathered}[/tex]

Hence the answer is 0.190 to the nearest thousandths

(b) In a fair cube all the numbers 1 to 6, have equal chances of rolling. So the probability of getting any number from 1 to 6 become

[tex]\begin{gathered} \frac{1}{total\text{ outcome}} \\ =\frac{1}{6} \\ =0.1666666666 \end{gathered}[/tex]

Hence the answer is 0.167 to the nearest thousandth

(c) If a cube is fair, then the larger the number of rolls, the greater the likelihood that the experimental probability will be close to the theoretical probability

The 3rd option is the answer

Lets start by copying the polynomial:

[tex](3x^8)^{15}+(m^{12})^6[/tex]

To simplify, we will need to use the following properties:

[tex]\begin{gathered} \mleft(a^cb^d\mright)^f=a^{cf}b^{df} \\ \mleft(a^b\mright)^c=a^{bc} \end{gathered}[/tex]

We will apply the first property to the first term and the second property to the second term:

[tex]\begin{gathered} (3x^8)^{15}+(m^{12})^6= \\ =3^{1\cdot15}x^{8\cdot15}+m^{12\cdot6}= \\ =3^{15}x^{120}+m^{72} \end{gathered}[/tex]

So, the simplification is:

[tex](3x^8)^{15}+(m^{12})^6=3^{15}x^{120}+m^{72}[/tex]

Jeff receives six dollars for each hour he babysits.

Let's set x for the number of hours:

Then, we can write the next function:

[tex]f(x)=6x[/tex]

We need to find the correct answers, using this function.

After 6 hours:

[tex]f(6)=6(6)=36[/tex]

After 8 hours:

[tex]f(8)=6(8)=48[/tex]

After 10 hours:

[tex]f(10)=6(10)=60[/tex]

After 12 hours:

[tex]f(12)=6(12)=72[/tex]

Looking at the results, the correct answer is the third table.

cost = $48.75

normal cost = $65

$65 ------------------------- 100%

$48.75 --------------------- x

x = (48.75 x 100) / 65

x = 75%

She saved 100 - 75 = 25%

The ratio of the areas of the circles is 9:49.

What is the relation between areas and the radius of the circle?

The area of a circle with respect to the radius of the circle is π[tex]r^{2}[/tex].

Now, it's given that the radii of the circles with A and B are 12in. and 28in.

Hence, the area of circle A is π[tex]12^{2}[/tex]= 144π

And the area of circle B is π[tex]28^{2}[/tex]=784π

Therefore the ratio of the area of a circle is 144π: 784π

Now, simplifying it, we get - 36:196 - 9:49

Hence the ratio of areas is 9: 49.

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400 out of the 900 students do not take the bus to school

Here, we want to know the number of students in the 7th grade that do not take a bus to school

Firstly, we need the fraction of the survey for students who do not take a bus

The total surveyed is 8+ 12 + 25 = 45

Out of this, (8+12 = 20) do not take the bus

The farction of the survey that do not take a bus is 20/45

So, out of the 900, the number that do not take a bus will be;

20/45 * 900 = 20 * 20 = 400 students

Answer:

B. 12 ft

Explanation:

We can see from the image that a right-angled triangle is formed by the ladder, the side of the house, and the ground.

We can use the Pythagorean Theorem to determine how far from the base of his house he should place the ladder as shown below;

[tex]\begin{gathered} 20^2=x^2+16^2 \\ 400=x^2+256 \\ 400-256=x^2 \\ 144=x^2 \\ \sqrt[]{144}=x \\ 12=x \\ x=12ft \end{gathered}[/tex]

the starting value is the value when f is on the inicial point or 0

1.

[tex]\begin{gathered} 2f-7 \\ 2(0)-7 \\ =-7 \end{gathered}[/tex]

2.

[tex]\begin{gathered} 6f+4 \\ 6(0)+4 \\ =4 \end{gathered}[/tex]

3.

[tex]\begin{gathered} 9f-2 \\ 9(0)-2 \\ =-2 \end{gathered}[/tex]

4.

[tex]\begin{gathered} 5f \\ 5(0) \\ =0 \end{gathered}[/tex]

5.

[tex]\begin{gathered} 3f+1 \\ 3(0)+1 \\ =1 \end{gathered}[/tex]

Growth value

to find growth value you do: f=1 - f=0

1.

[tex]\begin{gathered} 2f-7 \\ (2(1)-7)-(2(0)-7) \\ (-5)-(-7) \\ =2 \end{gathered}[/tex]

2.

[tex]\begin{gathered} 6f+4 \\ (6(1)+4)-(6(0)+4) \\ (10)-(4) \\ =6 \end{gathered}[/tex]

3.

[tex]\begin{gathered} 9f-2 \\ (9(1)-2)-(9(0)-2) \\ (7)-(-2) \\ =9 \end{gathered}[/tex]

4.

[tex]\begin{gathered} 5f \\ (5(1))-(5(0)) \\ 5-0 \\ =5 \end{gathered}[/tex]

5.

[tex]\begin{gathered} 3f+1 \\ (3(1)+1)-(3(0)+1) \\ 4-1 \\ =3 \end{gathered}[/tex]

On a geometric sequence, consecutive terms have a common ratio.

The formula for the n-th term a geometric sequence with first term a_1 an common ratio r is:

[tex]S(n)=a_1\cdot r^{n-1}[/tex]

In this case, we can see that the first term is 30,000. To find the common ratio, divide the value of the second term by the value of the first term:

[tex]\begin{gathered} r=\frac{a_2}{a_1} \\ =\frac{36,000}{30,000} \\ =1.2 \end{gathered}[/tex]

Substitute a_1=30,000 and r=1.2 to find the explicit formula for the geometric sequence:

[tex]S(n)=30,000\times1.2^{n-1}[/tex]

Problem: Olympia listened to some old records.

The first song lasted 2 minutes and 12 seconds, the

the second lasted 2 minutes and 16 seconds, the third 2

minutes and 18 seconds, and the fourth 3 minutes

and 4 seconds. Use mental math to determine the

total playing time for all four records.

Solution:

- Song 1 duration time: 2 minutes and 12 seconds.

- Song 2 duration time: 2 minutes and 16 seconds.

- Song 3 duration time: 2 minutes and 18 seconds.

- Song 4 duration time: 3 minutes and 4 seconds

Total playing time =

(2 minutes and 12 seconds)+ (2 minutes and 16 seconds)+(2 minutes and 18 seconds)+(3 minutes and 4 seconds).

This is equivalent to say:

Total playing time = 3x 2 minutes + 3 minutes + (12 + 16+18+4) seconds

= 6 minutes + 3 minutes + 50 seconds.

= 9 minutes + 50 seconds.

Then the correct answer is:

The total playing time is 9 minutes and 50 seconds.

Answer

The horizontal distance (x) when the ball hits the ground is 12.8 feet

SOLUTION

Problem Statement

The question tells us that a ball thrown from a height of 6 feet, is modeled by the following equation:

[tex]\begin{gathered} f(x)=-0.2x^2+2.1x+6 \\ \text{where,} \\ f(x)\text{ is the height of the ball} \\ x\text{ is the horizontal distance covered by the ball} \end{gathered}[/tex]

We are asked to find the horizontal distance covered by the ball before hitting the ground.

Method

- To find the horizontal distance covered before hitting the ground implies that the height of the ball after some horizontal distance covered must be zero.

- This is because for the ball to hit the ground, the height of the ball over the earth must be zero. This means that:

[tex]f(x)=0[/tex]

- Thus, to find the horizontal distance x when the ball hits the ground, then we need to equate the function of f(x) to zero and then solve for x.

That is:

[tex]\begin{gathered} \text{Solve,} \\ -0.2x^2+2.1x+6=0 \end{gathered}[/tex]

Implementation

[tex]\begin{gathered} -0.2x^2+2.1x+6=0 \\ \text{ We can solve this using the Quadratic Formula:} \\ \\ \text{The Quadratic formula is defined below:} \\ \text{Given the following Quadratic Equation:} \\ ax^2+bx+c=0 \\ \\ \text{The Quadratic formula is:} \\ x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \\ \\ \text{From the equation given, we can see that:} \\ a=-0.2,b=2.1,c=6 \\ \\ \therefore x=\frac{-2.1\pm\sqrt[]{2.1^2-4(-0.2)(6)}}{2(-0.2)} \\ \\ x=\frac{-2.1\pm3.0348}{-0.4} \\ \\ x=\frac{-2.1-3.0348}{-0.4}\text{ or }\frac{-2.1+3.0348}{-0.4} \\ \\ \\ \therefore x=12.837\text{ or }-2.337. \\ \\ \text{ Since a horizontal distance cannot be negative, we have that:} \\ x=12.837\approx12.8\text{ (to the nearest tenth)} \end{gathered}[/tex]

Final Answer

The horizontal distance (x) when the ball hits the ground is 12.8 feet

The area of the triangle = 1/2 b x h

A = 50 feet square

b = 5 feet

h = ?

50 = 1/2 x 5 x h

5/ 2 h = 50

5h = 50 x 2

5h = 100

h = 100/5

h= 20 feet

Answer:

Option C. 7x - 2y = -24

Explanations:

The equation of the line passing through the point (x₁, y₁) and parallel to the line y = mx + c is given by:

[tex]y-y_{1\text{ }}=m(x-x_1)[/tex]

The given equation is:

7x - 2y = 1

This can be simplied as:

[tex]\begin{gathered} 2y\text{ = 7x -1} \\ y\text{ = }\frac{7}{2}x\text{ - }\frac{1}{2} \end{gathered}[/tex]

The line is passing through the point (-2, 5)

The slope, m = 7/2

Substitute m = 7/2, x₁ = -2, and y₁ = 5 into the equation y - y₁ = m(x - x₁)

[tex]\begin{gathered} y\text{ - 5 = }\frac{7}{2}(x\text{ -(-2))} \\ y\text{ - 5 = }\frac{7}{2}(x\text{ + 2)} \\ 2(y\text{ - 5) = }7(x+2) \\ 2y\text{ - 10 = 7x + 14} \\ 7x\text{ - 2y = -10 - 14} \\ 7x\text{ - 2y = -24} \end{gathered}[/tex]

When applying Distributive Property to multiply (5j+7) and (-2j), the result will have two terms because there is one "j-term" and one "j²-term".

According to the distributive property of multiplication over addition,

Upon multiplying a number by the sum of more than two, the distributive property of multiplication over addition is applicable.

Here the two expressions given are -

(5j+7) and (-2j)

Now, multiplying these two expressions using Distributive Property of multiplication, we get

(5j+7) x (-2j)

= 5j x (-2j) + 7 x (-2j)

= -10j² - 14j

Thus, from the above expression of applying Distributive Property to multiply (5j+7) and (-2j), we have the result where there are two terms -

j-term

and, j²-term.

Therefore, option C is the correct answer because there will be a j-term and a j²-term when we apply the Distributive property to multiply (5j+7) and (-2j).

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Actually, you can't determine if they are congruent, you have one side and an angle, you need 1 more to know it

But if both down lines are parallel, then you know they have the same angle with the line you know.

Congruent by ASA

[tex](x)(4)+(x+5)(3)=Total\: distance[/tex]

1) The first thing to consider when tackling this question is that we are dealing with speed. And speed can be found this way:

[tex]speed=\frac{distance}{time}\Leftrightarrow d=st[/tex]

2) So, we can set our equation by calling x the distance. Since Mason took 4 hours and Isabella 3, we can start out writing:

[tex]\begin{gathered} dt_m+dt_I=Total\: distance \\ (x)t_M+(x+5)t_I=Total\: distance \\ (x)(4)+(x+5)(3)=Total\: distance \end{gathered}[/tex]

Note that the

We can simplify the expression 2(x - 1) + 3 (× + 2 ) ​ by following these steps:

1. Use the distributive property to get rid of the parentheses:

2(x - 1) + 3 (× + 2 )=2*x-2*1+3*x+3*2=2x-2+3x+6

2. Rewrite the like terms together:

2x-2+3x+6=2x+3x+6-2

3. Combine like terms:

2x+3x+6-2=(2+3)x+4=5x+4

Given:

Taylor weekly salary = $900

she earns 28% of the amount of her sales each week.

If she sells $1225 worth of

Answer:

C. (7,13)

Step-by-step explanation:

Since they are open circles, that means those values are not included, so we use parentheses instead of brackets.

When we round a number if the last digit is equal or greater than 5 we round up and if it is lower than 5 we round down. In this case we have:

Samuel: His plant is measuring 7.443 inches, the last digit is 3, therefore we must round down, which would be 7.44

Tash: Tasha's plant is measuring 7.455 inches, the last digit is 5, therefore we must round up. Which would be 7.46

Ben: His plant is measuring 7.446 inches. The last digit is 6, therefore we must round up. Which would be 7.45.

The answer is Ben.

If z varies inversely as w, we have:

[tex]z=k\frac{1}{w}[/tex]

Where k is constant

If, when z = 20, w = 0.7, we have:

[tex]\begin{gathered} 20=k\frac{1}{0.7} \\ k=20\cdot0.7 \\ k=14 \end{gathered}[/tex]

Therefore, for w = 20, we have:

[tex]\begin{gathered} z=14\frac{1}{20} \\ z=0.7 \end{gathered}[/tex]

At graph of Number 4: Let's first determine the equation of each line.

LINE 1: Let's use point A: (x1,y1) = (-5,-5) and point B: (x2,y2) = (-3,-4).

Getting the slope (m):

[tex]\text{ m = }\frac{y_2-y_1_{}}{x_2-x_1}[/tex][tex]\text{ m = }\frac{-4\text{ - (-5)}}{-3\text{ - (-5)}}\text{ = }\frac{-4\text{ + 5}}{-3\text{ + 5}}\text{ = }\frac{1}{2}[/tex]

Getting the y-intercept (b): Use m = 1/2 and x,y = -5,-5

[tex]\text{ y = mx + b}[/tex][tex]-5\text{ = (}\frac{1}{2})(-5)\text{ + b}[/tex][tex]\text{ -5 = }\frac{-5}{2}\text{ + b }\rightarrow\text{ b = -5 + }\frac{5}{2}[/tex][tex]\text{b = }\frac{\text{-10}}{2}\text{ + }\frac{5}{2}\text{ = }\frac{-10\text{ + 5}}{2}[/tex][tex]\text{ b = -}\frac{5}{2}[/tex]

Let's complete the equation: Substitute m and b in y = mx + b.

[tex]\text{ y = mx + b}[/tex][tex]\text{ y = (}\frac{1}{2})x\text{ + (-}\frac{5}{2})[/tex][tex]\text{ y = }\frac{1}{2}x\text{ - }\frac{5}{2}\text{ ; f(x) = }\frac{1}{2}x\text{ - }\frac{5}{2}[/tex]

LINE 2: Let's use point A: (x1,y1) = (-3,-2) and point B: (x2,y2) = (1,0).

Getting the slope (m):

[tex]\text{ m = }\frac{y_2-y_1}{x_2-x_1}[/tex][tex]\text{ m = }\frac{0-(-2)}{1-(-3)}=\frac{0\text{ + 2}}{1\text{ + 3}}\text{ = }\frac{2}{4}\text{ = }\frac{1}{2}[/tex]

Getting the y-intercept (b): Use m = 1/2 and x,y = 1,0

[tex]\text{ y = mx + b}[/tex][tex]\text{ 0 = (}\frac{1}{2})(1)\text{ + b}[/tex][tex]\text{ 0 = }\frac{1}{2}\text{ + b}[/tex][tex]\text{ b = -}\frac{1}{2}[/tex]

Let's complete the equation: Substitute m and b in y = mx + b.

[tex]\text{ y = mx + b}[/tex][tex]\text{ y = (}\frac{1}{2})x\text{ + }(-\frac{1}{2})[/tex][tex]\text{ y = }\frac{1}{2}x\text{ - }\frac{1}{2}\text{ ; f(x) = }\frac{1}{2}x\text{ - }\frac{1}{2}[/tex]

LINE 3: Let's use point A: (x1,y1) = (1,2) and point B: (x2,y2) = (3,3).

Getting the slope (m):

[tex]undefined[/tex]

To solve this question, we need to visualize the wheel. See below:

The motion f the pot is clearly a cosine motion, the minimum height is 0 instead of -12, so we can see that the cosine function has been shifted up by 12 units, so our equation is;

[tex]\begin{gathered} y=a\cos kx+12 \\ a\text{ is the amplitude of the height of the point above the centre only, so a =12} \\ y=12\cos kx+12 \\ k=\frac{2\pi}{\lambda} \\ \lambda=2\times\pi\times12=24\pi \\ so,k=\frac{2\pi}{24\pi}=\frac{1}{12} \\ \end{gathered}[/tex]

Thus, the complete equation is;

[tex]y=12\cos (\frac{1}{12}x)+12[/tex]

That is Option D

Let's solve each equation to find the right match.

[tex]2(e^{3t})=12[/tex]

First, we divide the equation by 2.

[tex]\begin{gathered} \frac{2(e^{3t})}{2}=\frac{12}{2} \\ e^{3t}=6 \end{gathered}[/tex]

Then, we apply logarithms on each side.

[tex]\begin{gathered} \log _e(e^{3t})=\log _e(6) \\ 3t=\log _e(6) \end{gathered}[/tex]

So the first expression matches the first answer choice.

The second equation is

[tex]12(e^{3t})=2[/tex]

First, we divide the equation by 12.

[tex]\begin{gathered} \frac{12(e^{3t})}{12}=\frac{2}{12} \\ e^{3t}=\frac{1}{6} \end{gathered}[/tex]

Then, we apply logarithms on each side.

[tex]undefined[/tex]

The area of the square sandbox must be 196 square feet.

So we want to recall the formula for the area of a square of side "L".

The area of the square is the product of the length of its side (L) times itself, which makes this:

[tex]\text{Area}=L\cdot L=L^2[/tex]

Then we set up an equation that states that the area of the square is 196 square feet, using the expression we have for the area in terms of the side length (L):

[tex]\begin{gathered} \text{Area}=196 \\ L^2=196 \end{gathered}[/tex]

And now we solve for the length L, by using the square root property:

[tex]\begin{gathered} L^2=196 \\ L=\sqrt{196} \\ L=14 \end{gathered}[/tex]

Therefore, if the area is 196 square feet, the length of the sides of the sandbox must be 14 feet

Then, the person needs to by 4 times 14 ft because the sandbox (square) has 4 sides,

That is: Mr Drew needs to buy 4 times 14 feet of wood, which gives a total of: 56 feet of wood.

Solution

Let the three consecutive even integers be h, (h+2) and (h+4)

The consecutive integers add up to 36, i.e

[tex]h+(h+2)+(h+4)=36[/tex]

Solve to find h

[tex]\begin{gathered} \text{Open the brackets} \\ h+(h+2)+(h+4)=36 \\ h+h+2+h+4=36 \\ \text{Collect like terms} \\ h+h+h+2+4=36 \\ 3h+6=36 \\ \text{Collect like terms} \\ 3h=36-6 \\ 3h=30 \\ \text{Divide both sides by 3} \\ \frac{3h}{3}=\frac{30}{3} \\ h=10 \end{gathered}[/tex]

The greatest of the integers is h + 4,

The value of the greatest integer is

[tex]\begin{gathered} h+4 \\ \text{Where }h=10 \\ h+4=10+4=14 \end{gathered}[/tex]

Hence, the greatest of the integers is 14

If you got 55.6% on a test and that test is worth 80%

So first let us calculate the actual percentage that you scored.

The actual percentage is

[tex]\begin{gathered} \frac{0.556}{0.80}=0.695 \\ 0.695\times100=69.5\% \end{gathered}[/tex]

So your actual score on the test is 69.5%

The grading system varies across countries and states

The traditional grading scale is given below

90% to 100% corresponds to A grade.

80% to 89% corresponds to B grade.

70% to 79% corresponds to C grade.

60% to 69% corresponds to D grade.

0% to 59% corresponds to F grade.

69.5% can be rounded off to 70%

Therefore, 69.5% corresponds to C grade.

(note: the above grade is not valid in case your school follows a different grading scale)

1. the value of BD is the same to AD so is 16

3. for the third we can calculate DN by pythagoras

NA=8, AD=16 and DN=?

but by pythagoras

[tex]\begin{gathered} NA^2+DN^{2^{^{}}}=AD^2 \\ 8^2+DN^2=16^2 \\ DN^2=16^2-8^2 \\ DN=\sqrt[]{16^2-8^2} \\ DN=8\sqrt[]{3}\approx13.86 \end{gathered}[/tex]

2.

We know that the man is 45 years old on August 27 and we are interested in determine the number of weeks. So we can do the following conversion:

[tex]45\text{years}\cdot\frac{52weeks}{1year}=2340weeks=2.34x10^3weeks[/tex]

We assume that for this case a year have 52 weeks