100 POINTS + BRAINLIEST
A teacher hires a coach for a school trip. The cost is worked out using the
formula C =
m
3 + 40, where C is the cost in pounds and m is the number of
miles the coach travels.
(a) Calculate how much it would cost to hire the coach to travel a distance of
42 miles.
b) If the cost of the hire is £75,how many miles does the coach travel?

Answer: a = 5 and k = 1/2

Step-by-step explanation:

We are given that the function is defined by:

f(x) = 16 + a(3^(kx))

We need to find the real numbers a and k such that f(0) = 21 and f(4) = 61. Using the given values of f(0) and f(4), we can form a system of two equations:

f(0) = 16 + a(3^(k(0))) = 21

f(4) = 16 + a(3^(k(4))) = 61

Simplifying the first equation, we get:

16 + a(3^0) = 21

16 + a = 21

a = 5

Substituting this value of a into the second equation, we get:

16 + 5(3^(4k)) = 61

5(3^(4k)) = 45

3^(4k) = 9

We know that 3^2 = 9, therefore

4k = 2

=> k = 2/4

=> k = 1/2

Therefore, the real numbers a and k that satisfy the given conditions are a = 5 and k = 1/2. So the function is:

f(x) = 16 + 5(3^(x/2))

The solution to the equation give us an approximate that x = 1.55.

To solve this equation, we can start by subtracting 4 from both sides to isolate the exponential term:

3.5^x + 4 - 4 = 11 - 4

This simplifies to:

3.5^x = 7

Next, we can take the logarithm of both sides with base 3.5 to eliminate the exponent:

log(3.5^x) = log(7)

Using the property of logarithms that log(a^b) = b*log(a), we can rewrite the left side as:

x*log(3.5) = log(7)

Now, we can solve for x by dividing both sides by log(3.5):

x = log(7) / log(3.5)

Plugging in the values, we get:

x = 0.84509804001 / 0.54406804435

x = 1.55329475566

x ≈ 1.55.

Answered question "Solve the equation 3.5^x + 4 = 11"

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Therefore, the basketball needs an additional 2,016π cubic centimeters of air to be fully inflated.

In physics and mathematics, volume refers to the amount of space occupied by a three-dimensional object or region of space. The volume of a solid object can be calculated using various formulas, depending on the shape of the object. The concept of volume is used in many areas of science and engineering, including physics, chemistry, architecture, and manufacturing. It is particularly important in fields such as fluid mechanics and thermodynamics, where the flow and behavior of liquids and gases are often analyzed in terms of their volume and the changes in volume that occur during various processes.

Here,

The volume of a fully inflated basketball is given by the formula V = (4/3)πr³, where r is the radius of the ball. In this case, the radius is 12 cm, so the volume of a fully inflated basketball is:

V = (4/3)π(12 cm)³

V = (4/3)π(1728 cm³)

V = 2,304π cm³

To find the volume of a basketball that is only inflated halfway, we need to find the radius of the half-inflated ball. Since the volume of a sphere is proportional to the cube of its radius, we can use the following proportion:

(Volume of half-inflated ball) / (Volume of fully inflated ball) = (Radius of half-inflated ball)³ / (Radius of fully inflated ball)³

Let Vh be the volume of the half-inflated ball, then:

Vh / 2,304π cm³ = (6 cm)³ / (12 cm)³

Simplifying this equation, we get:

Vh = (1/8) * 2,304π cm³

Vh = 288π cm³

Now we can find the additional volume of air needed to fully inflate the ball:

Volume of fully inflated ball - Volume of half-inflated ball = Additional volume of air needed

2,304π cm³ - 288π cm³ = 2,016π cm³

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To find the value of a6, we need to first determine the value of a2, a3, a4, and a5 using the given recurrence relation:

a1 = 6

a2 = 5a1 = 5(6) = 30

a3 = 5a2 = 5(30) = 150

a4 = 5a3 = 5(150) = 750

a5 = 5a4 = 5(750) = 3750

Now, we can find a6 using the recurrence relation:

a6 = 5a5 = 5(3750) = 18750

Therefore, the value of a6 is 18750.

Answer:

x = 81.5°

Step-by-step explanation:

the secant- tangent angle x is half the difference of the measures of the intercepted arcs.

the sum of the arcs in a circle = 360°, then

unknown intercepted arc = 360° - (215 + 93)° = 360° - 308° = 52°

then

x = [tex]\frac{1}{2}[/tex] (215 - 52)° = [tex]\frac{1}{2}[/tex] × 163° = 81.5°

The cosine of angle B is: cos(B) = AC / AB = 20/29

The Pythagorean Theorem, a well-known geometric theorem that states that the sum of the squares of the legs of a right-angled triangle is equal to the square of the hypotenuse (the opposite side of the right angle) - that is, in familiar algebraic notation. , a2 + b2 = c2 .

In a right triangle, the cosine of an angle is defined as the ratio of the adjacent side to its hypotenuse. In this case, angle B is the angle we are interested in, and the adjacent side is side AC. Using the Pythagorean theorem, we find the length of side AC:

AC² = AB²+ BC²

AC² = 29² - 21²

AC² = 400

AC = 20

Therefore, the cosine of angle B is:

cos(B) = AC / AB = 20/29

Other trigonometric ratios of angle B can be found using the following formulas:

sin(B) = BC / AB = 21/29

tan(B) = BC / AC = 21/20

csc(B) = AB / BC = 29 / 21

sec(B) = AB / AC = 29/20

bed (B) = AC / BC = 20 / 21  

Thus, the trigonometric ratios of angle B are:

sin(B) = 21/29

cos(B) = 20/29

tan(B) = 21/20

csc(B) = 29/21

sec(B) = 29/20

crib(B) = 20/21

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Part A: Number of servings  is  1.24 servings/person

Part A: To determine the number of complete servings of fruit salad, we need to divide the total amount of fruit salad Mrs. Harris made (31 cups) by the amount she wants each person to get (25 cup/person). Using the formula:

Number of servings = Total amount of fruit salad / Amount per serving

Number of servings = 31 cups / 25 cups/person

Number of servings = 1.24 servings/person

Since we cannot have a fraction of a serving, we round down to the nearest whole number, as we cannot serve a fraction of a fruit salad. Therefore, Mrs. Harris will be able to serve 1 complete serving of fruit salad with the amount she made.

art B:

If each cup of fruit salad contains 1/4 cup of fruit and each serving is 1/4 cup, then each serving contains 1 cup of fruit.

To calculate the number of bananas Mrs. Harris needs, we first need to determine how many cups of bananas are in 31 cups of fruit salad:

31 cups × 1/4 cup of banana per cup of fruit salad = 7.75 cups of bananas

Since each cup of banana contains 14 of a banana, we can multiply the number of cups of bananas by 14 to find the total number of bananas needed:

7.75 cups of bananas × 14 bananas per cup = 108.5 bananas

So Mrs. Harris needs to buy approximately 109 bananas for the fruit salad.

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Answer: a

Step-by-step explanation:

The vertical line at x=0 represents the boundary of the inequality, and the shaded region to the left of the line represents the solution set of the inequality, x < 0.

Explain the term inequality

Inequality refers to the unequal distribution of resources, opportunities, and rewards among individuals or groups within a society. It can be based on factors such as race, gender, income, education, and social status, leading to disparities in outcomes and experiences.

According to the given information

Start by simplifying the inequality using the distributive property and combining like terms:

4(x-1) - 2x < 6 - 5(x+2)

4x - 4 - 2x < 6 - 5x - 10

2x - 4 < -5x - 4

2x + 5x < 4 - 4

7x < 0

x < 0

So the solution to the inequality is x < 0.

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The volume of the suitcase is 35/8 cubic feet or approximately 4.375 cubic feet.

A rectangular prism is a three-dimensional solid object that has six faces, where each face is a rectangle. It is also called a rectangular parallelepiped. A rectangular prism is a type of prism because it has a constant cross section along its length.

A rectangular prism's volume V is determined by:

V = l × w × h

where l denotes the prism's length, w its width, and h its height. Here are the facts:

l = 2 3 ft

w = 1 1/2 ft

h = 1 1/4 ft

When we enter these values into the formula, we obtain:

V = (2 3 ft) × (1 1/2 ft) × (1 1/4 ft)

= (7/3 ft) × (3/2 ft) × (5/4 ft)

= 35/8 ft³

Therefore, the volume of the suitcase is 35/8 cubic feet or approximately 4.375 cubic feet.

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Answer:

No

Step-by-step explanation:

to determine if (- 5, 3 ) is a solution substitute the x- coordinate into both equations and if the value of y is equal to the y- coordinate then the point is a solution to the system.

y = x - 6 = - 5 - 6 = - 11 ≠ 3

y = - 2x - 7 = - 2(- 5) - 7 = 10 - 7 = 3

since both equations are not satisfied , then

(- 5, 3 ) is not a solution to the system

Answer:

To find the value of f(-1), we need to substitute -1 for x in the given function:

f(x) = 3x² + 1 - 1

f(-1) = 3(-1)² + 1 - 1

f(-1) = 3(1) + 0

f(-1) = 3

Therefore, f(-1) is equal to 3.

The polynomial function with the given properties is:[tex]f(x) = x^3 - 7x^2 + 9x - 17[/tex]

A polynomial function is a function that contains only non-negative integer powers or only positive integer exponents in the equation, such as a  quadratic equation, a cubic equation, etc. For example, 2x + 5 is a polynomial whose exponent is 1.

Since the polynomial has real coefficients and one of its zeros is complex, the complex conjugate of 4+ i, which is 4-i, must also be  zero. So the three zeros of the polynomial are -1, 4+i and 4-i.  To form a polynomial with these zeros, we start by writing  the factors of the polynomial:

(x + 1) (x - 4 - i) (x - 4 + i)

We can simplify this expression by multiplying  the factors:

[tex](x + 1) (x^2 - 8x + 17)[/tex]

Expanding this, we get:

[tex]x^3 - 7x^2+ 9x - 17[/tex]

Thus, the polynomial function with the given properties is:  

[tex]f(x) = x^3 - 7x^2 + 9x - 17[/tex]

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The polynomial function is p(x) = (-17/5)x³ + (34/5)x² + (33/5)x - 16

A polynomial function is a function that contains only non-negative integer powers or only positive integer exponents in the equation, such as a  quadratic equation, a cubic equation, etc. For example, 2x + 5 is a polynomial whose exponent is 1.

If -1 is a zero of the polynomial function, then x+1 is a factor of the polynomial. Similarly, if 4+i is a zero, then 4-i is also a zero, because complex roots always occur in conjugate pairs. Therefore, (x+1) and (x - 4 - i)(x - 4 + i) = (x - 4)² + 1 are factors of the polynomial. We can then construct the polynomial function by multiplying these factors:

p(x) = A(x+1)(x - 4 - i)(x - 4 + i)

where A is a constant that we need to determine, and p(x) is the desired third-degree polynomial function.

To determine A, we can use the y-intercept given in the problem. The y-intercept is the value of p(0), so:

p(0) = A(0+1)(0 - 4 - i)(0 - 4 + i) = A(17+4i)

But we also know that p(0) = -17, so:

-17 = A(17+4i)

Solving for A:

A = -17/(17+4i) = (-17/5) + (68/5)i

Therefore, the polynomial function we seek is:

p(x) = [(-17/5) + (68/5)i](x+1)(x - 4 - i)(x - 4 + i)

Expanding the product and simplifying, we get:

p(x) = (-17/5)x³ + (34/5)x² + (33/5)x - 16

This is a third-degree polynomial with only real coefficients, -1 and 4+i are two of the zeros, and the y-intercept is -17.

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Answer:

B and D

Step-by-step explanation:

A and C are linear because they are going in a straight line. E is going downwards, so it is decreasing. That leaves B and D, which have exponential growth.

Answer:

284.06

Step-by-step explanation:

To solve this expression using the order of operations (PEMDAS), we must first perform the operations inside the parentheses: 82-12 equals 70. Next, we must perform the multiplication and division from left to right: 107 divided by 70 equals approximately 1.53, and 1.53 times 122 equals approximately 186.06. Finally, we add 98 to get our answer. Therefore, the expression 98+107÷(82-12)x122 simplifies to approximately 284.06.

Answer:

284.06

Step-by-step explanation:

got it right on edge

The area of the shaded region is 73.65 cm².

We have,

The shaded region is a triangle.

Base = x

Height = y

Now,

Using the Pythagorean theorem,

y² = 12² + 10²

y² = 144 + 100

y² = 244

y = √244

And,

x² = 8² + 5²

x² = 64 + 25

x² = 89

x = √89

Now,

Area of the shaded region.

= 1/2 x √89 x √244

= 1/2 x 9.43 x 15.62

= 73.65 cm²

Thus,

The area of the shaded region is 73.65 cm².

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Answer:

x = 67.4°

y = 22.6°

Step-by-step explanation:

The tangent ratio is a trigonometric ratio that relates the ratio of the length of the side opposite an angle in a right triangle to the length of the side adjacent to that angle.

[tex]\boxed{\begin{minipage}{7 cm}\underline{Tangent trigonometric ratio} \\\\$\sf \tan(\theta)=\dfrac{O}{A}$\\\\where:\\ \phantom{ww}$\bullet$ $\theta$ is the angle. \\ \phantom{ww}$\bullet$ $\sf O$ is the side opposite the angle. \\\phantom{ww}$\bullet$ $\sf A$ is the side adjacent the angle.\\\end{minipage}}[/tex]

The side opposite angle x is 7.2, and the side adjacent angle x is 3.

Use the tangent trigonometric ratio to find the measure of angle x:

[tex]\begin{aligned}\tan(x)&=\dfrac{7.2}{3}\\x&=\tan^{-1}\left(\dfrac{7.2}{3}\right)\\x&=\vphantom{\dfrac12}67.380135...^{\circ}\\x&=67.4^{\circ}\;\sf (nearest\;tenth)\end{aligned}[/tex]

Therefore, x = 67.4°.

The side opposite angle y is 3, and the side adjacent angle y is 7.2.

Use the tangent trigonometric ratio to find the measure of angle y:

[tex]\begin{aligned}\tan(y)&=\dfrac{3}{7.2}\\y&=\tan^{-1}\left(\dfrac{3}{7.2}\right)\\y&=\vphantom{\dfrac12}22.6198649...^{\circ}\\y&=22.6^{\circ}\;\sf (nearest\;tenth)\end{aligned}[/tex]

Therefore, y = 22.6°.

Answer:

The odds that are less generous are 19:1, 14:1, and 15:1.

Step-by-step explanation:

You have to see which numbers are greater than 11.

The closest answer is 3xLn3 which is (B).

If k(x) = 3x, then f(x) can be expressed as the integral of k(x) with respect to x:

f(x) = ∫ k(x) dx = ∫ 3x dx = (3/2)x² + C

where C is the constant of integration.

To find f'(x), the derivative of f(x) with respect to x, we simply differentiate the expression for f(x):

f'(x) = d/dx [(3/2)x² + C] = 3x

A differential equation is a mathematical equation that relates a function and its derivatives. Specifically, a differential equation describes how a quantity changes as a function of its own rate of change.

Differential equations are commonly used in physics, engineering, and other sciences to model and analyze natural phenomena.

For example, the motion of a simple pendulum can be described using a differential equation that relates the position and velocity of the pendulum to its acceleration and the forces acting upon it.

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the given solution set {-5, -1} can be obtained by substituting either of the possible solutions into the absolute value equation and verifying that it holds.

what is an equation ?

In mathematics, an equation is a statement that asserts the equality of two mathematical expressions. An equation typically consists of two expressions separated by an equal sign, with the expression on the left side of the equal sign equal to the expression on the right side of the equal sign.

In the given question,

To write an absolute value equation that satisfies a given solution set, we need to consider the definition of absolute value. The absolute value of a number is its distance from zero the number line. Therefore, an absolute value equation can be written as follows:

|expression| = distance

where the expression is the quantity whose absolute value is being taken, and distance is a non-negative value representing the distance from zero. The equation is satisfied if and only if the expression has a distance from zero that matches the given distance.

For the solution set {-5, -1}, we can write the following absolute value equations for the given figures:

A number line with -5 and -1 marked as solutions:

|x + 3| = 2

This equation is satisfied when x = -5 or x = -1, because |-5 + 3| = 2 and |-1 + 3| = 2.

A number line with -5 and -1 equidistant from zero:

|x| = 5

This equation is satisfied when x = -5 or x = 5, because |-5| = 5 and |5| = 5.

Note that for both figures, the given solution set {-5, -1} can be obtained by substituting either of the possible solutions into the absolute value equation and verifying that it holds.

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Therefore, the degree of the given monomial is 11.

In algebra, a monomial is an expression consisting of a single term. A term can be a constant, a variable, or the product of a constant and one or more variables. In other words, a monomial is a polynomial with only one term.  The degree of a monomial is the sum of the exponents of its variables.

Here,

The degree of a monomial is the sum of the exponents of its variables.

For the monomial -3q⁴rs⁶, the degree would be:

4 + 1 + 6 = 11

Therefore, the degree of the monomial -3q⁴rs⁶ is 11.

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It should be noted that to determine the amplitude of a sinusoidal function from a graph:

Identify the highest point (peak) and the lowest point (valley) of the graph of the sinusoidal function.

Find the vertical distance between the peak and the midline of the graph (the line halfway between the highest and lowest points).

The amplitude of the sinusoidal function is half of this vertical distance.

Identify the highest point (peak) and the lowest point (valley) of the graph of the sinusoidal function.

Find the vertical coordinate of the midline of the graph (the line halfway between the highest and lowest points).

Write the equation of the midline as y = the vertical coordinate of the midline.

To determine the maximum value of a sinusoidal function from a graph:

Identify the highest point (peak) of the graph of the sinusoidal function.

The maximum value of the sinusoidal function is the vertical coordinate of this peak.

To determine the minimum value of a sinusoidal function from a graph:

Identify the lowest point (valley) of the graph of the sinusoidal function.

The minimum value of the sinusoidal function is the vertical coordinate of this valley.

To determine the period of a sinusoidal function from a graph:

Identify two consecutive peaks or valleys of the graph of the sinusoidal function.

Find the horizontal distance between these two points.

The period of the sinusoidal function is twice this horizontal distance.

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The  area of triangle ABC is 6cm²

A triangle is a three-sided polygon that consists of three edges and three vertices.  It is also known as the trigon and has special names such as the hypotenuse (opposite the right angle) and legs (the other two sides). It can also refer to a percussion instrument consisting of a rod of steel bent into the form of a triangle open at one angle.

The area of the triangle is given as

Area= 1/2absinC

But the CosB = Adj/Hypo

Cos60 = Adj/12

1/2 = Adj/12

corss and multiply to have

2A = 12

Therefore Adj = 6

Therefore base = 6*2 = 12

Applying the formula we have Area= 1/2acsinB

Area = 1/2* 12*12*Cos60

Area = 12*1/2 =6cm²

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Answer:

153.94 cm2

Step-by-step explanation:

Circle area = π * r² = π * 49 [cm²] ≈ 153.94 [cm²]

π ≈ 3.14159265 ≈ 3.14

d = r * 2 = 7 [cm] * 2 = 14 [cm]

MARK AS BRAINLIEST PLS I TOOK 3 HOURS THINKING ABOUT THIS

The variables I₁ and I₂ using the matrix algebra and using the Cramer's rule are I₁ = 1 and I₂ = 1

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

15I₁ + 5I₂ = 20

25I₁ + 5I₂ - 30 = 0

Rewrite as

15I₁ + 5I₂ = 20

25I₁ + 5I₂ = 30

Rewrite as

I₁     I₂

15    5    20

25   5    30

From the question, the matrix form is

AI = b

Ths matrix A from the above is

[tex]A = \left[\begin{array}{cc}15&5&25&5\end{array}\right][/tex]

Ths matrix B from the above is

[tex]B = \left[\begin{array}{c}20&30\end{array}\right][/tex]

And, we have the matrix I to be

[tex]I = \left[\begin{array}{c}I_1&I_2\end{array}\right][/tex]

Start by calculating the inverse of A from

[tex]A = \left[\begin{array}{cc}15&5&25&5\end{array}\right][/tex]

So, we have:

|A| = 15 * 5 - 5 * 25

|A| = -50

The inverse is

[tex]A^{-1} = -\frac{1}{50}\left[\begin{array}{cc}5&-5&-25&15\end{array}\right][/tex]

Recall that

AI = b

So, we have

[tex]I = -\frac{1}{50}\left[\begin{array}{cc}5&-5&-25&15\end{array}\right] * \left[\begin{array}{c}20&30\end{array}\right][/tex]

Evaluate the products

[tex]I = -\frac{1}{50}\left[\begin{array}{c}5 * 20 + -5 * 30&-25 * 20 + 15 *30\end{array}\right][/tex]

[tex]I = -\frac{1}{50}\left[\begin{array}{c}-50&-50\end{array}\right][/tex]

Evaluate

[tex]I = \left[\begin{array}{c}1&1\end{array}\right][/tex]

Recall that

[tex]I = \left[\begin{array}{c}I_1&I_2\end{array}\right][/tex]

So, we have

I₁ = 1 and I₂ = 1

Recall that the determinant of matrix A calculated in (a) is

|A| = -50

Replace the first column in A with b

So, we have

[tex]AI_1 = \left[\begin{array}{cc}20&5&30&5\end{array}\right][/tex]

Calculate the determinant

DI₁ = 20 * 5 - 30 * 5

DI₁ = -50

Replace the second column in A with b

So, we have

[tex]AI_2 = \left[\begin{array}{cc}15&20&25&30\end{array}\right][/tex]

Calculate the determinant

DI₂ = 15 * 30 - 20 * 25

DI₂ = -50

So, we have

I₁ = DI₁ / |A| = -50/-50 = 1

I₂ = DI₂ / |A| = -50/-50 = 1

So, we have

I₁ = 1 and I₂ = 1

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The volume of the rectangular prism in cubic centimeters is 3,600,000

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

Dimension = 120 centimeters by 2 meters by 1.5 meters

The volume of the rectangular prism is the product of the dimensions

i.e.

Volume = Length * width * height

Substitute the known values in the above equation, so, we have the following representation

Volume = 120 * 2 * 1.5

Convert to cm

Volume = 120 * 200 * 150

Evaluate

Volume = 3600000

Hence, the volume in cubic centimeters is 3,600,000

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The circumference of a child swimming pool is 6π.

What is circumference?

The circumference of a circle or ellipse in geometry is its perimeter. That is, if the circle were opened up and straightened out to a line segment, the circumference would be the length of the arc. The curve length around any closed figure is more often referred to as the perimeter.

Here, we have

Given: A child swimming pool that has a radius of 3 feet.

We have to find the circumference.

The circumference is diameter x π

The diameter is twice the radius

c = 2πr

c = 2π(3)

= 6π

Hence, the circumference of a child swimming pool is 6π.

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Answer:

Step-by-step explanation:

Dimensions of the rectangle are 6 feet and 12 feet.

Rectangle is a quadrilateral whose each pair of opposite sides are equal in length and each two consecutive sides are in right angle to each other. In rectangle one pair of equal opposite sides is called Length and another one is called Width.  

If length of a rectangle is L and width of rectangle is W then the perimeter of that rectangle will be = 2(L+W)

Let W be the rectangle's width.

Here according to question the rectangle is twice as long as it is wide.

So, length of rectangle = 2W

Perimeter will be = 2(W+2W) = 2*(3W) = 6W

So, according to question,

6W = 36

W = 36/6 = 6

Thus the width of the rectangle is = 6 feet.

Then the length = 2*6 = 12 feet.

Hence dimensions of the rectangle are 12 feet and 6 feet respectively.

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