The length of a rectangle is 6 cm longer than its width. if the perimeter of the rectangle is 36 cm , find its area.

The missing part of the unit rate in the given situation is 8 students per group. This is obtained from dividing the total number of students (40) by the total number of groups (5).

To find the missing part of the unit rate, we'll need to divide the total number of students (40) by the total number of groups (5). This is because a unit rate describes how many units of the first type of quantity corresponds to one unit of the second type of quantity.

So in this case, the unit rate would be 40 ÷ 5 = 8. Therefore, the missing part of the unit rate is 8 students per group.

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The equation of the parabola that models the movement of the football is y = -2x² + 5x + 12. The time at which the ball hits the ground is x = 4 seconds.

The quadratic equation is defined as a function containing the highest power of a variable is two.

a. We know that the general form of the quadratic equation is y = ax² + bx + c, where a, b, and c are constants.

We are given three points that the football passes through: (1, 15), (2, 14), and (3, 9).

We can substitute these points into the quadratic equation and solve for a, b, and c.

For the first point, we have: 15 = a(1)² + b(1) + c

For the second point, we have: 14 = a(2)² + b(2) + c

For the third point, we have: 9 = a(3)² + b(3) + c

Solving this system of equations using elimination gives us:

a = -2

b = 5

c = 12

Therefore, the equation of the parabola that models the movement of the football is y = -2x² + 5x + 12.

b. To find the time at which the ball hits the ground, we can set the value of y equal to 0 and solve for x.

y = -2x² + 5x + 12

0 = -2x² + 5x + 12

-12 = -2x² + 5x

-2x² + 5x - 12 = 0

We can use the quadratic formula to solve for x:

x = (-5 +/- √(5² - 4(-2)(-12)))/(2(-2))

x = (-5 +/- √(25 + 96))/(-4)

x = (-5 +/- √(121))/(-4)

x = (-5 +/- 11)/(-4)

Therefore, x = 4 or x = -3/2.

The time at which the ball hits the ground is x = 4 seconds.

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Answer:

its c

Step-by-step explanation:

To find the inverse of f(x) = x² + 7, replace f(x) with y and solve for x, resulting in the inverse function f–1(x) = √(x - 7), considering the domain x ≥ 7.

To find the inverse function of f(x) = x² + 7, we must first replace f(x) with y, giving us y = x² + 7. To find the inverse, we solve for x. We start by subtracting 7 from both sides to get y - 7 = x². Taking the square root of both sides, we get x = ±√(y - 7).

Since the square root has two values (positive and negative), we choose the one that matches the domain and range of the original function. If f(x) is defined for all nonnegative values of x, the inverse would be f–1(x) = √(x - 7). If f(x) includes negative values of x, then we would consider both, resulting in two different functions.

However, because we're dealing with f(x) that implies only one output for each input without restrictions, we consider the principal square root. Thus, the equation for the inverse function is f–1(x) = √(x - 7), considering the domain where x is greater than or equal to 7.

Answer:

The solution to the given inequality is (-∞,5].

Step-by-step explanation:

An inequality is a relation which makes a non-equal comparison between two numbers.

Given : 7-3(x+5)+8x≤17

=7-3(x+5)+8x ≤ 17

= 7- 3x - 15 + 8x ≤ 17

= -8 + 5x ≤ 17

Adding 8 on both sides:

= -8 +8+ 5x ≤ 17+8

= 5x ≤ 25

x ≤ 5

The solution to the inequality : (-∞,5].

The equation for the function f would be; f(x) = -1.75x + 18.75.

An equation is an expression that shows the relationship between two or more numbers and variables.

Function is a type of relation, or rule, that maps one input to specific output.

Given that f is a linear function such that f(5) = 10 and f(9) = 3, then

Consider that the function of x would be;

f(x) = -1.75x + 18.75

Now plug x = 5 in the given function, if it satisfy the function

f(5) = -1.75(5) + 18.75

f (5) =10

Again plug x = 9 in the given function, if it satisfy the function

f(9) = -1.75(9) + 18.75

f(9) =3

Therefore, the equation for the function f would be; f(x) = -1.75x + 18.75.

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Final answer:

The average speed of the car is 86.67 km/h.

Explanation:

The average speed of a car can be calculated by dividing the total distance traveled by the total time taken. In this case, the car traveled 85 km from town A to town B and then 45 km from town B to town C, for a total distance of 85 km + 45 km = 130 km. The total trip took 1.5 hours. To find the average speed, divide the total distance by the total time: 130 km / 1.5 h = 86.67 km/h.

The polynomial equation of least degree with the roots -1, 3, and (+/-)3i is x⁴ - 2x³ + 10x² + 2x - 27 = 0.

We have to find the polynomial equation of least degree with the given roots -1, 3, and (+/-)3i. As complex roots occur in conjugate pairs, if 3i is a root, then -3i must also be a root. Therefore, our polynomial will be of degree 4 since it has four roots in total.

The polynomial of least degree with roots x1, x2, x3, x4 can be expressed as:

(x - x1)(x - x2)(x - x3)(x - x4) = 0

Plugging in the roots given, we get:

(x + 1)(x - 3)(x - 3i)(x + 3i) = 0

To simplify this expression, let's first multiply the complex factors:

(x - 3i)(x + 3i) = x² - (3i)² = x² + 9

Now, multiply the remaining real factors with the result we just got:

(x + 1)(x - 3)(x² + 9) = 0

Expanding this, we obtain the polynomial equation:

x⁴ - 2x³ + 10x² + 2x - 27 = 0

This is the simplified equation of the polynomial with roots -1, 3, and (+/-)3i.

A) (-6,0) =  -6-3, 0+4 so it moves to (-9,4)

B) (3,-4) = 3-3, -4+4 so it moves to (0,0)

The correct answer is option 3. The equation for the line of reflection that maps the trapezoid onto itself is y = 3.

To determine the line of reflection for a trapezoid that maps onto itself, we need to identify the axis of symmetry.

We can analyze each one:

1. x = 0 represents a vertical line of reflection. If the trapezoid were reflected over this line, the left and right sides would be swapped. This would not map the trapezoid onto itself unless the trapezoid is symmetric about the y-axis, which is not specified.

2. x = 3 also represents a vertical line of reflection. This would reflect the trapezoid over a vertical line three units to the right of the y-axis. Again, this would not map the trapezoid onto itself unless the trapezoid is symmetric about the line x = 3, which is not specified.

3. y = 3 represents a horizontal line of reflection three units above the x-axis. If the trapezoid has its bases parallel to the x-axis and the midline of the trapezoid is at y = 3, then reflecting over this line would indeed map the trapezoid onto itself. This is because the top half of the trapezoid would be a mirror image of the bottom half across the line y = 3.

4. y = 0 represents a horizontal line of reflection along the x-axis. Reflecting the trapezoid over the x-axis would flip it upside down, which would not map the trapezoid onto itself unless the trapezoid is symmetric about the x-axis, which is not typical for a trapezoid.

The complete question is:

What is the equation for the line of reflection that maps the trapezoid onto itself?

1. x = 0

2. x = 3

3. y = 3

4. y = 0

Answer: (a,a)

Step-by-step explanation:

Given: A  right isosceles triangle which is placed on the coordinate plane such that one of the legs of the triangle lies along the x-axis and the other leg is parallel to the y-axis.

From the given figure one vertex is on origin(0,0) and second on x -axis as (a,0) .ii.e. one side of the triangle is on the x axis with length 'a' and the side parallel to the y axis should perpendicular to the x axis with length 'a'.(∵ two sides are equal in a right isosceles triangle)

So the coordinates should she assign to the third vertex of the right isosceles triangle is (a,a).[as the point lie above the point (a,0), so its x abscissa should be same]

The third vertex of a right isosceles triangle where one leg lies on the x-axis and the second leg is parallel to the y-axis would have equal x and y coordinates, assuming the second vertex is at (a,0). Therefore, the third vertex would be at (a,a), for any real numerical value of 'a'. This makes use of key concepts in coordinate geometry.

In the context of mathematics, specifically coordinate geometry, Mei can assign the third vertex of this right isosceles triangle to any point that is equally distant from both the x and y axes. Assuming the other two vertices of the triangle are at the origin (0,0) and on the x-axis (a,0), then the coordinates of the third vertex would be (a,a).

Right isosceles triangle is a special type of triangle where two sides are equal in length and the angle between them is 90 degrees. Since the legs of the triangle are along the x and y axes, this leads to the third vertex being on the line y=x.

A coordinate proof involves using the properties and definition of the geometric figures in the coordinate plane. It involves calculations with the coordinates of the vertices of the figures. In the case of coordinate geometry, it's essential to understand the placement of points on the x-y plane.

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The maximum age of a fossil that could be dated using 40K is approximately 5.36 billion years.

Potassium-40 (40K) has a half-life of 1.25 billion years.

The minimum detectable amount for dating using 40K is 0.1%. If a dinosaur is dated using 40K to be 67 million years old, we can calculate the maximum age of a fossil that could be dated using 40K.

To find the maximum age, we can set up a proportion using the half-life of 40K:

(67 million years) / (1.25 billion years) = (x years) / (100%)

Solving for x, we get:

x = (67 million years) * (100%) / (1.25 billion years)

Calculating this, we find that the maximum age of a fossil that could be dated using 40K is approximately 5.36 billion years.

double every 6 months 

in 6 months there would be 250*2 = 500 crickets

 in 1 year there would be 500*2 = 1000 crickets

in 1.5 years there would be 1000 *2 = 2000 crickets

 in 2 years there would be 2000 *2 = 4000 crickets

Answer:

amount evaporated =  0.635 liters

Step-by-step explanation:

The scientist initially had 5 liters of water in a container . After evaporation took place he had only 4.365 liters of water remaining in the container.

The amount of water that evaporated can be computed when you remove the amount of water remaining after evaporation from the  initial amount before evaporation.

Mathematically,  

amount evaporated = initial amount before evaporation - amount remaining after evaporation

initial amount before evaporation = 5 liters

amount remaining after evaporation = 4.365 liters

amount evaporated =  5 - 4.365

amount evaporated =  0.635 liters

Final answer:

There can be a total of 11,175 two-person conversations at the party.

Explanation:

In a party with 150 people, the number of two-person conversations that can occur can be calculated using the formula for combinations. Since each conversation involves two people, we need to choose 2 people from the total of 150. The formula for combinations is:

C(n, r) = n! / (r!(n-r)!)

Where n is the total number of people and r is the number of people we need to choose. Plugging in the values, we have:

C(150, 2) = 150! / (2!(150-2)!)

C(150, 2) = (150 ×149) / (2 ×1)

C(150, 2) = 11175

Therefore, there can be a total of 11,175 two-person conversations at the party.

Final answer:

At a party with 150 people, there can be 11175 unique two-person conversations, calculated using the combination formula 150C2.

Explanation:

The question is asking how many two-person conversations can occur at a party with 150 people. This is a classic combination problem where we need to calculate the number of unique pairs that can be made from 150 people. Since the order in which the pair is chosen does not matter, we use the combination formula nCr = n! / (r!(n-r)!), where 'n' is the total number of people and 'r' is the group size (in this case, 2 for a two-person conversation).

For 150 people, the calculation is:

150C2 = 150! / (2!(150-2)!) = 150! / (2! x 148!) = (150 x 149) / (2 x 1) = 11175.

Therefore, there can be 11175 unique two-person conversations at a party with 150 people.