Adam got 56 out of 84 correct in his test. What fraction of the marks did he get correct

Answer:

1) [tex]x=\dfrac{5}{2}[/tex]

2)  [tex]t_1=-1[/tex] and [tex]t_2=6[/tex]

3) (0,12)

4) [tex]\left(\dfrac{5}{2},\dfrac{49}{2}\right)[/tex]

Step-by-step explanation:

The function [tex]h(t) =-2t^2 + 10t + 12[/tex] represents the height (h) of the ball after t seconds.

Find the vertex of the parabola:

[tex]t_v=-\dfrac{b}{2a}=-\dfrac{10}{2\cdot (-2)}=\dfrac{5}{2}\\ \\h_v=h\left(\dfrac{5}{2}\right)=-2\cdot\left(\dfrac{5}{2}\right)^2+10\cdot \dfrac{5}{2}+12=-2\cdot \dfrac{25}{4}+25+12=-\dfrac{25}{2}+37=\dfrac{49}{2}[/tex]

Hence, the vertex of the function is at point [tex]\left(\dfrac{5}{2},\dfrac{49}{2}\right).[/tex]

The axis of symmetry of the function is vertical line which passes through the vertex, so its equation is

[tex]x=\dfrac{5}{2}[/tex]

To find y-intercept, equat t to 0 and find h:

[tex]h=-2\cdot 0^2+10\cdot 0+12=12[/tex]

Hence, y-intercept is at point (0,12)

To find the roots of the quadratic finction, equate h to 0 and solve the equation for t:

[tex]h=0\Rightarrow -2t^2+10t+12=0\\ \\t^2-5t-6=0\ [\text{Divided by -2}]\\ \\D=(-5)^2-4\cdot 1\cdot (-6)=25+24=49\\ \\t_{1,2}=\dfrac{-(-5)\pm \sqrt{49}}{2\cdot 1}=\dfrac{5\pm 7}{2}=6,\ -1[/tex]

Therefore, two roots are [tex]t_1=-1[/tex] and [tex]t_2=6[/tex]

Answer:

See the explanation.

Step-by-step explanation:

Given function [tex]f(x)=x^3-x^2-6x+2[/tex]

And the interval [tex][0,3][/tex]

According to Rolle's Theorem

Let [tex]f(x)[/tex] be differentiable on the open interval [tex](a,b)[/tex] and continuous on the closed interval [tex][a,b][/tex]. Then if [tex]f(a)=f(b)[/tex], then there is at least one point [tex]x\ in\ (a,b)[/tex] where [tex]f'(x)=0[/tex].

So,

[tex]f(0)=0^3-0^2-6\times0+2\\\\f(0)=2\\\\Similarly\\\\f(3)=3^3-3^2-6\times3+2\\\\f(3)=27-9-18+2\\\\f(3)=2\\\\We\ can\ see\ f(0)=f(3)\\\\Now,\\\\f'(x)=\frac{d}{dx}(x^3-x^2-6x+2)\\\\f'(x)=3x^2-2x-6\\\\put\ f'(x)=0\\\\3x^2-2x-6=0\\[/tex]

We will find the value of [tex]x[/tex] for which [tex]f'(x)[/tex] became zero.

[tex]If\ ax^2+bx+c=0\\\\Then,\ x_{1}=\frac{-b+\sqrt{b^2-4ac}}{2a}\\ \\And\ x_{2}=\frac{-b-\sqrt{b^2-4ac}}{2a}\\\\f'(x)=3x^2-2x+6=0\\a=3,\ b=-2,\ c=6\\\\\ x_{1}=\frac{-(-2)+\sqrt{(-2)^2-4\times3\times(-6)}}{2\times3}\\\\x_{1}=\frac{2+\sqrt{76}}{6}=\frac{2+2\sqrt{19}}{6}\\\\x_{1}=\frac{1+\sqrt{19}}{3}\\\\x_{2}=\frac{-(-2)-\sqrt{(-2)^2-4\times3\times(-6)}}{2\times3}\\\\x_{1}=\frac{2-\sqrt{76}}{6}=\frac{2-2\sqrt{19}}{6}\\\\x_{2}=\frac{1-\sqrt{19}}{3}[/tex]

We can see

[tex]x_{1}=\frac{1+\sqrt{19}}{3}=1.786\\\\and\ 1.786\ is\ in\ [0,3][/tex]

There is at least one point [tex]1.786\ in\ (0,3)[/tex] where [tex]f'(x)=0[/tex].

Rolle's theorem requires a function to be continuous on [a, b], differentiable on (a, b), and have f(a) = f(b). After verifying these conditions are met for f(x) = x^3 - x^2 - 6x + 2, we look for a point in (0, 3) where f'(x) = 0. We cannot find such a point in the interval, suggesting an error in the application of the theorem.

To verify Rolle's theorem for the function f(x) = x^3 - x^2 - 6x + 2 in the interval [0,3], we need to ensure that the function meets the criteria stated by the theorem:

The function f must be continuous on the closed interval [a, b].

The function f must be differentiable on the open interval (a, b).

The function f must satisfy f(a) = f(b).

f(x) = x^3 - x^2 - 6x + 2 is a polynomial, which is continuous and differentiable everywhere. Therefore, it satisfies the first two conditions of Rolle's theorem on any interval, including [0,3].

Now we need to check the third condition:

f(0) = (0)^3 - (0)^2 - 6(0) + 2 = 2
f(3) = (3)^3 - (3)^2 - 6(3) + 2 = 2

Since f(0) = f(3), the third condition is also met. Hence, Rolle's theorem applies, and there must be at least one c in (0, 3) such that f'(c) = 0.

To find c, we take the derivative of f(x):

f'(x) = 3x^2 - 2x - 6

Set f'(x) equal to zero and solve for x:

3x^2 - 2x - 6 = 0
Factors into (3x + 2)(x - 3) = 0
So x can be either x = -2/3 or x = 3. Since -2/3 is not in our interval, the only possibility in the interval (0, 3) is at x = 3. However, in this case, the point x = 3 is an endpoint, hence it's not within the open interval (0, 3).

Thus, we need to conclude that there has been a mistake, as x = 3 does not satisfy the condition for Rolle's theorem within the open interval (0, 3).

Answer:

111/40

Step-by-step explanation:

3 2/5=17/5

17/5-5/8=111/40

Answer:

x = 35°

Step-by-step explanation:

The question is as following

cos x = sin(20 + x)°  and  0° < x  < 90°  ,   find X?

==============================================

cos x = sin(20 + x)°

sin and cos are co-functions,

which means that: cos x = cos [90 - (20 + x)]

∴ x = 70 - x

∴ 2x = 70

∴ x = 35°

======================

Note: cos θ = sin ( 90 - θ )

Answer:

x = 35°

Step-by-step explanation:

The question is as following

cos x = sin(20 + x)°  and  0° < x  < 90°  ,   find X?

==============================================

cos x = sin(20 + x)°

sin and cos are co-functions,

which means that: cos x = cos [90 - (20 + x)]

∴ x = 70 - x

∴ 2x = 70

∴ x = 35°

======================

Note: cos θ = sin ( 90 - θ )

Answer:

The probability that carbon emissions from the factory are within the permissible level and the test predicts the opposite to be true is 19.338% (Rounding to the next thousandth place)

Step-by-step explanation:

1. Let's review the information provided to us to answer the question correctly:

Probability that carbon emissions from the company’s factory exceed the permissible level = 35% = 0.35

Accuracy of the test of emissions level = 85% = 0.85

2. The probability that carbon emissions from the factory are within the permissible level and the test predicts the opposite to be true is?

These two events, carbon emissions from the company’s factory and the accuracy of the test are independent events, therefore:

Probability that carbon emissions from the factory are within the permissible level = 1 - 0.35 = 0.65

Probability that the test predicts the opposite to be true = 0..35 * 0.85 = 0.2975 (The opposite is that the carbon emissions from the company exceed the permissible level)

Probability that carbon emissions from the factory are within the permissible level and the test predicts the opposite to be true is:

0.65 * 0.2975 = 0.193375

The probability that carbon emissions from the factory are within the permissible level and the test predicts the opposite to be true is 19.338% (Rounding to the next thousandth place)

Final answer:

The probability that the emissions are within the permissible level and the test incorrectly predicts the opposite is 0.098 or 9.75% after rounding to the thousandth place.

Explanation:

The probability that carbon emissions from the factory are within the permissible level is the complement of the probability that they exceed the permissible level, which is 100% - 35% = 65% or 0.65. Given that the test has an accuracy of 85%, the probability that the test incorrectly predicts the emissions to be over the permissible level when they are not (a false positive) is 15% or 0.15. Therefore, the probability that emissions are within the permissible level and the test incorrectly predicts the opposite is the product of the two probabilities: 0.65 * 0.15 = 0.0975 or 9.75% when expressed as a percentage. To provide the answer to the thousandth place as required, we express this as 0.098.

Answer

given,

352 + 116

we have to break 116 and use it on open number line

breaking of number 116

116 = 100 + 10 + 6

And addition will be shown in the number line attached below

now,

352 + 100 = 452

452 + 10 = 462

462 + 6 = 468

we know

sum of

352 + 116 = 468

Answer:166.4 or 166

Step-by-step explanation:

Answer:

x = -0.9

Step-by-step explanation:

You need to isolate x in order to find x so,

-3.4 - x = -2 1/2 or -2.5

-3.4 - x = -2.5

Add 3.4 to -2.5

-3.4 - x = -2.5

+3.4       +3.4

-x = 0.9

x CANNOT be negative so divide -1 to both sides

-x/-1 = 0.9/-1

x = -0.9

Checking answer

-3.4 - (-0.9) = -2.5

-3.4 + 0.9 = -2.5

-2.5 = -2.5

1. Slopes:

  - The first line has a negative slope of -1/4.

  - The second line has a positive slope of 1/4.

2. Y-intercepts:

  - The first line intersects the y-axis at y = 2.

  - The second line intersects the y-axis at y = -2.

To compare these lines, let's first rewrite both equations in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.

1. For the first equation, [tex]\( Y = -\frac{1}{4}x + 2 \)[/tex]:

  - Slope (m₁) = -1/4

  - Y-intercept (b₁) = 2

2. For the second equation, [tex]\( 2y = \frac{1}{2}x - 4 \)[/tex]:

  - We divide both sides by 2 to isolate y:

    [tex]\( y = \frac{1}{4}x - 2 \)[/tex]

  - Slope (m₂) = 1/4

  - Y-intercept (b₂) = -2

Now, let's compare:

1. Slopes:

  - The first line has a negative slope of -1/4.

  - The second line has a positive slope of 1/4.

2. Y-intercepts:

  - The first line intersects the y-axis at y = 2.

  - The second line intersects the y-axis at y = -2.

Answer:

A

Step-by-step explanation:

That seems like it would be the correct equation.

Answer:

a

Step-by-step explanation:

it is the tell how many days

Note: As you missed to identify what we have to find in this question. But, after a little research, I am able to find that we had to find the Mode for the data given in your question. So, I am assuming we have to calculate the the Mode. Hopefully, it would clear your concept regarding this topic.

Answer:

The mode of the data = 75

Step-by-step explanation:

Lets visualize the given data in a table to show the frequency distribution:

Daily expenditure on milk (in Rs)           Number of households

                 0-30                                                           5

                30-60                                                          6        

                60-90                                                          9

                90-120                                                         6

                120-150                                                        4

Here the maximum frequency is 9.

So, modal class is 60-90.

As the formula to calculate the mode:

[tex]Mode = l_{1} + h (\frac{f_{1}-f_{0}}{2f_{1}-f_{0}-f_{2}} )[/tex]

Here, the maximum

[tex]l_{1} =60, f_{1} =9, f_{0}=6, f_{0}=6, h=30[/tex]

[tex]l=[/tex] is the lower limit of the class

[tex]f_{1} =[/tex] is the frequency of the modal class

[tex]f_{0} =[/tex] is the frequency of the previous modal class

[tex]f_{2} =[/tex] is the frequency of the next previous modal class

[tex]l=[/tex] is the class size

So,

[tex]Mode = l_{1} + h (\frac{f_{1}-f_{0}}{2f_{1}-f_{0}-f_{2}} )[/tex]

[tex]Mode = 60 + 30 (\frac{9-6}{2(9)-6-6} )[/tex]

[tex]Mode = 60 + \frac{(30)(3)}{6}[/tex]

[tex]Mode = 60 + 15=75[/tex]

∴ The mode of the data = 75

Keywords: mode, frequency distribution

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The correct answer is:

The light bulb consumes 1056 watt-hours per day, calculated by dividing the total watt-hours by the total number of days.

To find out how many watt-hours the light bulb consumes per day, we need to divide the total watt-hours consumed by the total number of days.

Given:

- Total watt-hours consumed = 5280 watt-hours

- Total time = 5 days and 12 hours

First, let's convert the total time to hours:

[tex]\[ \text{Total time} = 5 \times 24 \text{ hours} + 12 \text{ hours} = 120 + 12 = 132 \text{ hours} \][/tex]

Now, we can calculate the watt-hours consumed per day:

[tex]\[ \text{Watt-hours per day} = \frac{\text{Total watt-hours}}{\text{Total number of days}} \]\[ \text{Watt-hours per day} = \frac{5280 \text{ watt-hours}}{5 \text{ days}} \]\[ \text{Watt-hours per day} = \frac{5280}{5} \text{ watt-hours} \]\[ \text{Watt-hours per day} = 1056 \text{ watt-hours} \][/tex]

So, the light bulb consumes 1056 watt-hours per day.

Answer:

D. 35

Step-by-step explanation:

[tex]20 \times \frac{7}{4} = 35[/tex]

Good morning,

Answer:

D. 35

Step-by-step explanation:

it Has the ratio of 4 to 7 then

it Has the ratio of 4×5 to 7×5 then

it Has the ratio of 20 to 35.

:)

Answer:

4x

Step-by-step explanation:

Answer:

Length of rectangle= 13

Step-by-step explanation:

Area= Length × width

91= length × 7

length= 91 ÷7 = 13

Answer:

(-6,16), also can be written as x=-6, y=16.

Step-by-step explanation:

By 'sub method' I assume you mean substitution of one solved equation into another. Let's begin by solving x+y=10 for y.

[tex]x+y=10[/tex]

-x on both sides

[tex]y=10-x[/tex]

Now we have an equation that tells us y=10-x. We can now replace 'y' in the other equation with this new definition of y, that being 10-x.

[tex]2x+y=4[/tex]

Substitute 10-x for y

[tex]2x+(10-x)=4[/tex]

When can remove the parentheses as they are unnecessary.

[tex]2x+10-x=4[/tex]

Now we solve for x. We can first simplify, by combining all like terms on each side, those being 2x and -x on the left side. 2x+-x=x.

[tex]x+10=4[/tex]

Now we remove 10 from both sides, to isolate x.

[tex]x=4-10[/tex]

We can again simplify, 4-10=-6

[tex]x=-6[/tex]

Now we have the x coordinate of the point that solves our set of equations. We can plug this number back as an x value of either equation to get the y value for the solution. In this example I will use x+y=10, as it is a simpler equation.

[tex]x+y=10[/tex]

Substitute x for -6.

[tex]-6+y=10[/tex]

Isolate y by adding 6 to both sides, cancelling out the -6 on the left side.

[tex]y=10+6[/tex]

Simplify 10+6=16.

[tex]y=16[/tex]

Now we have an x value, -6, and a y value, 16. These are the x and y values of our solution, therefore the solution is (-6,16).

This is the solution using the sub method. There are other ways to solve this question (although the answer, if done correctly, will always be the same), including plotting both lines on a graph, as shown in the attached image.

Answer:

The correct answer is x = 3

Complete question and statement:

Given the equilateral triangle Δ ABC in the graph attached,

AB = 6x - 3

AC = 3x + 6

What is the value of x?

Source: Previous question that can be found at brainly

Step-by-step explanation:

Let's recall that an equilateral triangle has its three sides equal. therefore:

AB = AC

6x - 3 = 3x + 6

3x = 9 (Common terms)

x = 9/3 (Dividing by 3 at both sides)

x = 3 ⇒ AB = 15 units, AC = 15 units, BC = 15 units

Answer:

Step-by-step explanation:

when x=35

y=-35+175=140

Answer:225.28

Step-by-step explanation: 98 become 100 and 22528 stays the same.

It will take [tex]4\frac{2}{7}\ hours[/tex] to cover same distance.

Step-by-step explanation:

Given,

Time period = 5 hours

Speed = 60 km/h

Distance = Speed * Time

Distance = 60*5 = 300 km

Now,

Distance = 300km

Speed = 70 km/h

Distance = Speed * Time

Time = [tex]\frac{Distance}{Speed}[/tex]

[tex]Time=\frac{300}{70}\\\\Time=\frac{30}{7}\\\\Time= 4\frac{2}{7}\ hours[/tex]

It will take [tex]4\frac{2}{7}\ hours[/tex] to cover same distance.

Keywords: speed, distance

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Yes, 4/19 is a rational number because it can be expressed as a fraction, where both the numerator and the denominator are integers and the denominator is not zero.

Yes, 4/19 is a rational number. In mathematics, a rational number is defined as a number that can be expressed as the quotient or fraction p/q of two integers, where the denominator q is not equal to zero. Here, 4 and 19 are both integers, and 19 (the denominator) is not zero, so 4/19 meets the criteria for being a rational number. An example of an irrational number would be the square root of 2, because it cannot be exactly expressed as a fraction.

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Answer:

[tex]\mathbb{R}-\{0\}[/tex]

[tex](-\infty,0)\ U\ (0,+\infty)[/tex]

Step-by-step explanation:

The domain of a Function

Given a real function f(x), the domain of f is made of all the values x can take, such that f exists. The function given in the question is

[tex]\displaystyle f(x)=-\frac{1}{x^2}-4[/tex]

Finding the domain of a function is not possible by giving x every possible value and check if f exists in all of them. It's better to find the values where f does NOT exist and exclude those values from the real numbers.

Since f is a rational function, we know the denominator cannot be 0 because the division by 0 is not defined, so we use the denominator to find the values of x to exclude from the domain.

We set

[tex]x^2=0[/tex]

Or equivalently

x=0

The domain of f can be written as

[tex]\mathbb{R}-\{0\}[/tex]

Or also

[tex](-\infty,0)\ U\ (0,+\infty)[/tex]

ans is 9/4 .................

The width (W) of the prism is 16 cm.

The length (L) of the prism is 16 cm.

The height (H) of the prism is 10 cm.

The formula for the surface area of any right prism with width W, length L, and height H is the following:

A = 2 (WL + LH + WH)

Substituting using our given values changes the equation to:

A = 2 (16*16 + 16*10 + 16*10)

= 2 (256 + 160 + 160) = 2 (576)

= 1152

Thus, the surface area is 1152 cm².

Let me know if you need any clarifications, thanks!

Answer:

1152

Step-by-step explanation:

A table is given containing the values of x and y.

The equation of the line is modeled by the values in the table.

As we can see in the table, we get to know that;

The y value is increased by 5 whereas the value of x is increased by 2.

So, we will put the numbers over each other for slope.

Therefore, slope=5/2

Now, we will put the slope value in the model equation;

y=mx+b

y=5/2x+b

Now, in the place of y and x we will be putting (2,1) in their place;

1=5/2(2)+b

b=1-5=-4

So, the final equation we get as;

y=5/2x-4

Hence, the equation of the line that is modeled by the values in the table shown is y=5/2x-4.

equation, statement of equality between two expressions consisting of variables and/or numbers. In essence, equations are questions, and the development of mathematics has been driven by attempts to find answers to those questions in a systematic way.

A mathematical equation which represents the relationship of two expressions on either side of the sign. It mostly has one variable and equal to symbol. Example: 2x – 4 = 2.

An equation is a statement that says that the value of two mathematical expressions is equal. In simple words, an equation says that two things are equal. It is denoted by the equal to sign '='. Example of an Equation: 8+2= 12-2. The above equation says that the left side of the equation is equal to the right side.

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Answer:

The x-intercept is the ordered pair (9,0)

Step-by-step explanation:

we know that

The x-intercept is the value of x when the value of y is equal to zero

so

The x-intercept is a ordered pair with a y-coordinate equal to zero

therefore

In this problem

The x-intercept is the ordered pair (9,0)

The x-intercept from the list of ordered pairs (8,10), (3,-4), (0,8), (4,-3), (9,0) is 9, as it is the x-value of the pair where the y-coordinate is zero.

To find the x-intercept from a list of ordered pairs, you need to look for the pair where the y-coordinate is zero. The x-intercept is the x-value in this pair.

From the list of ordered pairs given in the question, which are (8,10), (3,-4), (0,8), (4,-3), (9,0), we identify that the x-intercept is in the pair where y is equal to 0.

Thus, the x-intercept from the list is 9, as it appears in the ordered pair (9,0).

Answer:

in polynomial form

Step-by-step explanation:

if you have any question then reply me

The area of the rectangle can be found by multiplying its width by its height, which is a polynomial expression in this case.

To find the area of a rectangle, we multiply its width by its height. In this case, the width of the rectangle is y3+6y and the height is 2y3+5. So, the area of the rectangle is:

A = width × height

A = (y3+6y) × (2y3+5)

A = 2y^6 + 5y^3 + 12y^4 + 30y

This is the area of the entire rectangle, expressed as a polynomial in standard form.

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Answer:

Angle bisector

Step-by-step explanation:

Find the measure of the angle COA. By angle addition postulate,

[tex]m\angle COX=m\angle AOX+m\angle COA[/tex]

From the diagram,

[tex]m\angle COX=80^{\circ}\\ \\m\angle AOX=40^{\circ},[/tex]

then

[tex]80^{\circ}=m\angle COA+40^{\circ}\\ \\m\angle COA=80^{\circ}-40^{\circ}=40^{\circ}[/tex]

Find the measure of the angle BOA. By angle addition postulate,

[tex]m\angle BOX=m\angle AOX+m\angle BOA[/tex]

From the diagram,

[tex]m\angle BOX=60^{\circ}\\ \\m\angle AOX=40^{\circ},[/tex]

then

[tex]60^{\circ}=m\angle BOA+40^{\circ}\\ \\m\angle BOA=60^{\circ}-40^{\circ}=20^{\circ}[/tex]

Find the measure of the angle COB. By angle addition postulate,

[tex]m\angle COX=m\angle BOX+m\angle COB[/tex]

From the diagram,

[tex]m\angle BOX=60^{\circ}\\ \\m\angle COX=80^{\circ},[/tex]

then

[tex]80^{\circ}=m\angle COB+60^{\circ}\\ \\m\angle COB=80^{\circ}-60^{\circ}=20^{\circ}[/tex]

This means, the measures of angles COB and BOA are the same and are equal half the measure of angle COA, so angles COB and BOA are congruent. This means, the ray OB is the angle bisector of angle COA

Answer:

Option B is correct.

[tex]x=-8[/tex]

Step-by-step explanation:

Given:

The given equation is.

[tex]7x=-56[/tex]

Solve above equation for value of x.

[tex]7x=-56[/tex]

[tex]x=-\frac{56}{7}[/tex]

[tex]x=-8[/tex]

Therefore. the value of [tex]x=-8[/tex]

The calculated cost of spraying the rectangular field is $846

Calculating the cost of spraying the rectangular field

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

Dimension = 720 m by 500 m

Unit cost = $23.50 per hectare

The area of the field is calculated as

Area = 720 m * 500 m

So, we have

Area = 360000 square meters

Converted to hectares, we have

Area = 36 hectares

So, we have

Cost = $23.50 per hectare * 36 hectares

Evaluate

Cost = $846

Hence, the cost of spraying the rectangular field is $846