Is 61 composite or prime

Answer:

(a-6)(a-1)(a+6)

Final answer:

The lowest common denominator for the fractions [tex]6/(a^2-7a+6)[/tex] and [tex]3/(a^2-36)[/tex] is (a-6)(a-1)(a+6).

Explanation:

To find the lowest common denominator for the fractions [tex]6/(a^2-7a+6)[/tex] and [tex]3/(a^2-36),[/tex] we need to determine the least common multiple of the denominators.

The denominator of the first fraction can be factored as (a-6)(a-1) and the denominator of the second fraction can be factored as (a-6)(a+6).

The common factors are (a-6) and (a-1)(a+6).

Therefore, the lowest common denominator is (a-6)(a-1)(a+6).

The forest with an area of 5000 km² decreases by 5.75% annually. After 14 years, the forest area will be approximately 2,182 km².

The interest that is computed using both the principal and the interest that has accrued during the previous period is called compound interest.

The compound interest formula is:

[tex]A = P(1 + \frac{r}{n})^{nt}[/tex]

Where,

A is compounded amount

P is principal amoutn

r is the rate of interest

n is the number of times it is compounded

Given that,

Initial forest area  (P): 5000 km²

Rate of decrease per year: 5.75% or 0.0575

Number of years (n): 14

Since the area is decreasing,

Therefore r = -0.0575

Plugging in the values in the compound interest formula,

[tex]A = 5000 \times (1 - 0.0575)^{14} \\\\\approx 2,182 \text{ square km}[/tex]

Hence,

After 14 years, the forest area would be approximately 2,182 km².

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Answer

part 1 = c/a

part 2 = b/a

Sine is the ratio of opposite to hypotenuse respect to as a given angle while tangent is the ratio of opposite to adjacent lengths of a given angle.

Part 1

Sine = opposite/hypotenuse

SinB = b/a

Tangent = opposite/adjacent

TanC = c/b

SinB × TanC = b/a×c/b

                       = c/a

Part 2

SinC = c/a

TanB = b/c

SinC×TanB= c/a×b/c

                   = b/a

The ratio b/a is found by dividing both sides of the equation 5a = 20b by 5a, resulting in b/a being B) 1/4.

The question asks us to solve for the ratio b/a given the equation 5a = 20b. To find b/a, we can divide both sides of the equation by 5a:

From this, we can see that b/a must be 1/4, since 4 multiplied by 1/4 equals 1. Therefore, the correct answer is B. 1/4.

we know that

A relationship between two variables, x, and y, represent a direct variation if it can be expressed in the form [tex]y/x=k[/tex] or [tex]y=kx[/tex]

In this problem we have

[tex]p=20.8\\q=31.2[/tex]

Find the value of k

[tex]k=q/p[/tex]

Substitute the values

[tex]k=31.2/20.8[/tex]

[tex]k=1.5[/tex]

The linear equation is

[tex]q=1.5p[/tex]

For [tex]q=15.3[/tex]

Find the value of p

Substitute in the linear equation and solve for p

[tex]15.3=1.5p[/tex]

[tex]p=15.3/1.5=10.2[/tex]

therefore

the answer is the option a

[tex]10.2[/tex]

Answer:

The value of x is,  [tex]\frac{3}{4}[/tex]

Explanation:

Given:

The exponential equation [tex]9^{2x} =27[/tex]        ....[1]

Exponential Identities:

[tex](x^a) =x^{ab}[/tex]

we can write equation [1] as;

[tex](3^2)^{2x} = 3^3[/tex]     [ [tex]3 \times 3= 9 , 3 \times 3 \times 3 = 27 [/tex] we can rewrite as [tex]3^2 =9 , 3^3 =27[/tex]]

by using above identities, we have

[tex](3)^{4x} = 3^3[/tex] then,

[tex]4x =3[/tex]          [ if [tex]x^a =x^b[/tex] then a =b]

Simplify:

[tex]x = \frac{3}{4}[/tex]

To find the area of a circle segment, you need to know the radius and central angle of the segment. Use the formula Area of Circle Segment = ((central angle/360) x πr²) - 0.5r²sin(central angle) to calculate the area.

To calculate the area of the segment or shaded area of a circle, you must first need to know the radius of the circle and the central angle of the segment, as these will be necessary for the calculation. The formula for calculating the area of a segment in a circle is Area of Circle Segment = ((central angle/360) x πr²) - 0.5r²sin(central angle).

For instance, if the circle has a radius of 5 cm and the central angle of the segment is 60 degrees, the area is ((60/360) x π5²) - 0.5 x 5²sin(60), which will give the result when evaluated.

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Answer: 4n + 12 = 3d

Step-by-step explanation:

The equation representing "If the numerator of a fraction is increased by 3, the fraction becomes 3/4" is 4(x + 3) = 3y. This, alongside the second condition of the denominator decreased by 7 leading to the fraction becoming 1, provides two equations that can be solved for the original fraction.

To represent the situation "If the numerator of a fraction is increased by 3, the fraction becomes 3/4," we can use the variable x for the numerator and y for the denominator of the original fraction. Therefore, the equation would be (x + 3)/y = 3/4. Applying the cross-product rule, we multiply each side of the fraction by the denominator on the other side, leading to 4(x + 3) = 3y. Simplified, this equation is 4x + 12 = 3y, which helps us solve for the original fraction along with the other given condition that reducing the denominator by 7 makes the fraction equal to 1. So, the second condition can be written as x/(y - 7) = 1. These two equations together can be solved simultaneously to find the values of x and y.

Answer:

0

Step-by-step explanation:

Final answer:

To find the volume of a cylindrical can, one must apply the formula for volume of a cylinder, V = πr²h, with the given dimensions. The can's volume is approximately 603.19 cubic centimeters or milliliters, translating to about 0.603 liters.

Explanation:

The student is asking about the volume of a can, which is the measure of how much space it occupies or, in this context, how much liquid it can hold. To find the volume of a cylindrical can, we can use the formula V = πr²h, where V is the volume, r is the radius, and h is the height.

Since the can is 12 cm high and 8 cm wide (which is the diameter), the radius would be 4 cm (which is half of the diameter). Substituting these values into the formula gives us V = π(4cm)²(12 cm), which simplifies to V = π × 16 cm² × 12 cm). When calculated, the volume of the can is approximately 603.19 cubic centimeters.

Since the student may also be interested in capacity in terms of common liquid measurements, it is useful to know that 1 cubic centimeter is equivalent to 1 milliliter.

Thus, the can would be able to hold approximately 603.19 milliliters. Given that 1000 milliliters make up 1 liter, the can's capacity would be roughly 0.603 liters, which is a little over half a liter.

A triangle cannot have two right angles because the sum of the interior angles in any triangle is equal to two right angles. Having two right angles would necessitate that the third angle also be a right angle, forming a straight line instead of a triangle.

Proof that a Triangle Cannot Have Two Right Angles

The notion that a triangle cannot have two right angles is fundamentally rooted in the geometry axiom stating that the sum of the interior angles in any triangle is equal to two right angles (or 180 degrees). If a triangle were to have two right angles, the third angle would also have to be a right angle to satisfy the sum of 180 degrees. However, if the third angle is a right angle, this contradicts the definition of a triangle being a three-sided polygon with three angles that add up to 180 degrees; with three right angles, the figure would no longer be a triangle, as the three lines AC, CD, and BD would line up to form a straight line, eliminating the closed polygon structure of a triangle.

If we consider the work of notable mathematicians such as Legendre and Dehn, we find substantial evidence supporting the statement that the sum of the interior angles of a triangle cannot be greater than two right angles. Legendre's attempts to prove this led to understanding the consistency of triangle angle sums across all triangles; that is, if one triangle's angles added up to two right angles, it would be the same for all triangles. Furthermore, Dehn's hypothesis indicated that without parallel lines, the sum of the angles of a triangle is greater than two right angles.

Overall, every triangle must have at least two acute angles, and any attempt to form a triangle with two right angles would result in a shape that does not adhere to the fundamental properties of a triangle.

The laser will cut the archway at height of 3.5 feet and 4 feet (Option C).

A linear equation exists an equation in which the highest power of the variable stands always 1. It exists also known as a one-degree equation. The standard form of a linear equation in one variable exists in the form Ax + B = 0. Here, x is a variable, A exists as a coefficient and B is constant.

A parabola exists as a plane curve that stands mirror-symmetrical and is approximately U-shaped. It fits several superficially various mathematical descriptions, which can all be proved to determine exactly the same curves.

Refer to the following figure:

The blue line represents eqn of archway: y = -0.5x^2 + 3x, and green line represent eqn of laser: -0.5x + 2.42y = 7.65.

Now to find out the points at which laser cuts archway, we need to equate both the eqns.

[tex]-0.5x^{2} +3x=\dfrac{7.65+0.5x}{2.42}[/tex]

[tex]-1.21x^{2} +7.26x=7.65+0.5x[/tex]

[tex]-1.21x^{2} +6.76x-7.65[/tex]=0

On solving the quadratic eqns, we get x =3.5 and 4 (approximate)

Therefore, point A and B are 3.5 and 4 feet respectively.

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Final answer:

The sentence 'The product of Matt's score and 6 is 96' translates to the equation m × 6 = 96. By dividing both sides by 6, the value of Matt's score, denoted as m, is found to be 16.

Explanation:

To translate the given sentence into an equation, you start by identifying the mathematical operation described. The sentence states 'The product of Matt's score and 6 is 96'. The word 'product' indicates multiplication. Next, you use the variable m to represent Matt's score.

The equation that represents this situation is:

m × 6 = 96

To solve for m, you divide both sides of the equation by 6:

m = 96 ÷ 6

m = 16

Therefore, Matt's score is 16.