Evie was doing a science fair project by surveying her biology class. She found that of the 30 students in the class, 15 had brown hair and 17 had blue eyes and 6 had neither brown hair nor blues eyes. Determine the number of students who had brown hair and blue eyes.

To solve the equation, let’s translate the given information into an algebraic equation. The solution to the equation is x = -18.

To solve the equation, let’s translate the given information into an algebraic equation. Let the number be represented by 'x'. The quotient of the number and 3 is x/3. The problem states that four less than the quotient of a number and 3 is -10, so we can write the equation as:

x/3 - 4 = -10

To solve for x, we can start by adding 4 to both sides:

x/3 = -6

Next, we can multiply both sides of the equation by 3 to isolate x:

x = -18

Therefore, the solution to the equation is x = -18.

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What applies here is one of the laws of the factorization of polynomials, called the factor theorem and it states that:

For a polynomial f(x) , if for any value a,  f(a) =0  then (x-a) is factor of f(x)

Example:

Consider the polynomial

[tex]f(x) = x^{3} - 3x^{2} - 8x+4[/tex]

For  a =3,

[tex]f(a) = (3)^{3} - 3(3)^{2} - 8(3)+24[/tex]

= 27-27-24+24 = 0

f (3)= 0

which means (x-3) is a factor of f(x)

Applying the above rule to the question:

if (x + 3) is a factor of a polynomial f(x), then f(-3) = 0

note that (x+3) can also be written  as (x- (-3)).

To find the inverse of the function y = x² - 7, interchange x and y to get x = y² - 7, then solve for y to get the inverse function y = (x + 7), taking into account both the positive and negative square roots.

To find the inverse of the function y = x² − 7, we need to swap the x and y variables and then solve for y. Here's a step-by-step process:

This results in the inverse function y = ±√(x + 7), but typically, we only take the principal square root for the inverse function, which would be y = √(x + 7) assuming x ≥ -7.

Answer:

C. t = –4h + 21

Step-by-step explanation:

We know that when h = 0, t = 21°F. Replacing h = 0 in  equation A and D we get:

A.

t = 21h + 4

t = 21(0) + 4

t = 4

D.

t = –21h + 4

t = –21(0) + 4

t = 4

So, none of them are correct.

The temperature drops 4 degrees every hour, this means that for h = 1 then t = 21 - 4 = 17. Replacing h = 1 in  equation B and C we get:

B.

t = 4h + 21

t = 4(1) + 21

t = 25

C.

t = –4h + 21

t = –4(1) + 21

t = 17

In consequence, C is the correct option.

Answer: The correct option are A, B and D.

Explanation:

The law of sine states that,

[tex]\frac{\sin A}{a} =\frac{\sin B}{b}=\frac{\sin C}{c} [/tex]

Where A, B, C are interior angles of the triangle and a, b, c are sides opposite these angles respectively as shown in below figure.

Since we need the combination of two angles and one side or two sides and one angle.

The Law of sine is useful for AAS and SSA type problems.

Reason for correct option:

In  option A three sides are known but no angle is not given, therefore the SSS problem can not be solved by Law of sine and the option A is correct.

In option B a side is known and two inclined angle on that line are known. But to use Law of sine we want the line and angle which in not inclined on that line, therefore the ASA problem can not be solved by Law of sine and the option B is correct.

In option D two sides and their inclined angle is known. But to use Law of sine we want the side and angle which in not inclined on that line, therefore the SAS problem can not be solved by Law of sine and the option D is correct.

Reason for incorrect option:

In option C, the two consecutive angles are given and a side which makes the second angle with base side, therefore the first angle is opposite to the given side, so the law of sine can be used for AAS problems.

Therefore, option C is incorrect.

Answer:  The correct option is (B) [tex]f>5.[/tex]

Step-by-step explanation:  Given that Blake has more than five friends.

We are to select the inequality that describes the number of Blake's friends.

The number of Blake's friends is represented by f.

According to the given information, we have

the inequality that describes the number of Blake's friends is given by

[tex]f>5.[/tex]

Thus, (B) is the correct option.

Answer:

The answer is x=-3.

Step-by-step explanation:

I'm not sure how to word it, but I did this question on khan, and got this answer and it was right.

Final answer:

The dimensions of each corral should be 40 meters by 10 meters.

Explanation:

To find the dimensions of the rectangular corrals, we can set up an equation using the perimeter and area of the enclosed space. Let's call the length of one corral x and the width y. The perimeter of the two corrals is 2x + 2y, which equals 100 meters. The area of the enclosed space is xy, which equals 350 square meters.

Using these equations, we can solve for x and y. Rearranging the perimeter equation, we get x = 50 - y. Substituting this into the area equation, we have (50 - y)y = 350.

Simplifying the equation, we get y^2 - 50y + 350 = 0. This is a quadratic equation that can be factored as (y - 35)(y - 10) = 0. Therefore, the possible values for y are 35 and 10.

Since we are looking for positive values for the dimensions, we choose the values y = 10 and x = 50 - y = 50 - 10 = 40. Therefore, the dimensions of each corral should be 40 meters by 10 meters.

To find the dimensions of the rectangular corrals, we can set up a system of equations. By solving the system of equations, we find that the dimensions of the rectangular corrals can be either 25 meters by 14 meters or 7 meters by 50 meters to obtain a total area of 350 square meters.

To find the dimensions of the rectangular corrals, we can set up a system of equations. Let x represent the width of one corral and y represent the length. Since the rancher wants to enclose a total of 350 square meters, we have the equation xy = 350. The perimeter of each corral is 2x + y, so the total amount of fencing used would be 4x + 2y.

Given that the total fencing available is 100 meters, we can set up the equation 4x + 2y = 100. Now we can solve the system of equations:

By substituting the value of y from the first equation into the second equation, we can solve for x. After finding the value of x, we can substitute it back into the first equation to find the corresponding value of y. The solutions will give us the dimensions of the rectangular corrals.

Let's solve the system of equations:

The solutions for x are x = 25 and x = 7. Plugging these values back into the equation xy = 350, we find that the corresponding values for y are y = 14 and y = 50, respectively. Therefore, the dimensions of the rectangular corrals can be either 25 meters by 14 meters or 7 meters by 50 meters to obtain a total area of 350 square meters.

The points that are solutions to the system of inequalities x + y 5 + 2 y > 2 below are (5, 2),  (3, 6), (1, 1). Options B, C, A, and D. For is mathematically given as

Generally, inequalities are simply defined as  the relationship between two non-equal expressions using a symbol like "not equal to," "greater than," or "less than."

In conclusion, The points (5, 2), and (2, 5), (1, 1) are the ones that are the solutions to the system of inequalities x + y 5 + 2 y > 2 below (3, 6).

For (5, 2),

5 + 2* 5 + 2 *2 > 2

19>2

For  (2, 5)

2 + 5* 5 + 2*5 > 2

37>2

For (3, 6).

3 + 6*5 + 2*6 > 2

45>2

For (1, 1)

1 + 1* 5 + 2*1 > 2

8>2

For (1, -1)

1 + -1* 5 + 2 *-1 > 2

-6<2

For (2, -5)

2 + -5* 5 + 2*-5 > 2

−33<2

Read more about inequalities  

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Answer:

m∠3=115°.

Step-by-step explanation:

It is given from the figure that line q is parallel to s and r is the transversal.

Since, m∠6 and m∠8 forms a linear pair as they are on the straight line r, therefore using the linear pair property, we have

m∠6+m∠8=180°

⇒[tex]2x-5+x+5=180^{\circ}[/tex]

⇒[tex]3x=180^{\circ}[/tex]

⇒[tex]x=60^{\circ}[/tex]

Thus, the measure of ∠6 is [tex]2x-5=2(60)-5=120-5=115^{\circ}[/tex]

Now, m∠3=m∠6=115° as both m∠3 and m∠6 forms the alternate interior angle pair.

Therefore, the measure of m∠3=115°.

D. Y=-(3+x^2)

A. Y=2x-x^2

Answer:

The answers are a) y=2x-x^2 and d) y=-(3+x^2) and in the attached file are the graphs.

Step-by-step explanation:

Quadratic equations are those where the exponent of the unknown term is squared, that is, the unknown is elevated to exponent 2. They have the general form of a trinomial:

ax2 + bx + c = 0

where a, b and c are real numbers and are called coefficients. Thus, a is the coefficient of x2, b is the term or coefficient of x and c is the independent term.

Answer:

value of [tex](f o g)(1)[/tex] is, 17

Step-by-step explanation:

We have to find the [tex](f o g)(1)[/tex]

Using the given tables;

[tex](f o g)(1) = f(g(1))[/tex]   ......[1]

At x = 1

g(1) = 6

Substitute this in [1] we have;

[tex](f o g)(1) = f(6)[/tex]

At x = 6

f(6) = 17

then;

[tex](f o g)(1) = 17[/tex]

Therefore, the value of [tex](f o g)(1)[/tex] is, 17

The length of chord CD is 5 unit.

The line segment connecting any two locations on a circle's circumference is referred to as the chord of the circle. It should be emphasised that the diameter is the circle's longest chord, which runs through its centre.

We have,

As, the distance to both chords are 5.70 unit.

and,  both chords are 90 degrees from the line then the chords are identical.

As, the length of Chord AB = 5 unit then the length of chord CD =5.

Thus, the length of chord CD is 5 unit.

Learn more about Chord here:

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To estimate the number of grains of sand that fit in a 5-gallon bucket, we can calculate the volume of the bucket and the volume of a grain of sand. By assuming the grains are approximately the same size and shape, we can use the formula for the volume of a sphere to estimate the number of grains.

To answer this question, we need to make some assumptions. Let's assume that the grains of sand are roughly the same size and shape. We can estimate the number of grains of sand that fit in a 5-gallon bucket by considering the volume of the bucket and the volume of a grain of sand. The volume of a grain of sand can be approximated as a sphere.

Using the formula for the volume of a sphere (V = (4/3)πr³), we can find the volume of a grain of sand. If the grain of sand has sides that are 1.0 mm long, the radius of the sphere would be 0.5 mm (half of the side length).

Now, we can calculate the volume of the 5-gallon bucket, convert it to cubic millimeters, and divide it by the volume of a grain of sand to estimate the number of grains that fit in the bucket.

A 5-gallon bucket can hold approximately 36 million grains of sand.

To estimate the number of grains of sand that can fit into a 5-gallon bucket, we need to start with the volume of the bucket and an average sand grain.

A standard 5-gallon bucket is approximately 18.927 liters (since 1 gallon = 3.78541 liters).

An average grain of sand has a diameter ranging from 0.063 mm to 2 mm. We'll use an average grain size of 1 mm for our calculations. The volume V of a sphere (which we can use to approximate a sand grain) is given by the formula:

V = 4/3 π r³

For a grain of sand with a 1mm diameter, the radius r is 0.5 mm or 0.0005 meters. Therefore:

V = 4/3 π (0.0005)³ ≈ 5.24 x 10-10 m³

The volume of the 5-gallon bucket in cubic meters is:

18.927 liters = 0.018927 m³

Dividing the volume of the bucket by the volume of a single grain of sand gives us:

Number of grains = 0.018927 / 5.24 x 10-10 ≈ 3.613 x 107 grains

Therefore, approximately 36 million grains of sand can fit into a 5-gallon bucket.