It takes 40 min for a bus to travel the 36 miles from Framingham to Worcester. A car traveling from Worcester to Framingham moves 1.5 times as fast as the bus. If the car and the bus start moving towards each other simultaneously, after how many minutes will they meet?

The number of people George can feed is 6.

The unitary method is a technique for solving a problem by first finding the value of a single unit, and then finding the necessary value by multiplying the single unit value.

Given that, George bought 4 submarine sandwiches for a birthday party.

Each person will eat 2/3 of a sandwich

Number of people George can feed

= 4÷2/3

= 4×3/2

= 6

Therefore, the number of people George can feed is 6.

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Answer: No

Step-by-step explanation: It does not go through the origin (0,0) in a straight line. Also, it has a y-intercept, meaning it doesn't go through the origin (0,0) and goes through the line that is up and down; vertically.

To find the absolute minimum and absolute maximum values of the function f(x) = x - ln 8x on the interval [1/2, 2], we find the critical points and evaluate the function at the endpoints and critical points. The absolute minimum value is f(1/2) = 1/2 - ln 4 and the absolute maximum value is f(2) = 2 - ln 16.

To find the absolute minimum and absolute maximum values of the function f(x) = x - ln 8x on the given interval [1/2, 2], we need to find the critical points and the endpoints of the interval. First, we find the derivative of the function: f'(x) = 1 - 1/x. Then, we solve the equation f'(x) = 1 - 1/x = 0 to find the critical points. The critical point is x = 1. Next, we evaluate the function at the critical point and the endpoints of the interval to determine the absolute minimum and absolute maximum values.

1. Evaluating the function at the critical point: f(1) = 1 - ln 8
2. Evaluating the function at the endpoint x = 1/2: f(1/2) = 1/2 - ln 4
3. Evaluating the function at the endpoint x = 2: f(2) = 2 - ln 16

The absolute minimum and absolute maximum values of the function on the given interval are:

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x = 0

y = 1 + 2x Substitute x = 0

y = 1 + 2(0) Any number times 0 is 0

y = 1 + 0 Simplify

y = 1

I hope this could help.

The solution to the system of equations is x = 1, y = 2, and z = -1.

The given system of equations is:

1. x + y - 2z = 5

2. -x + 2y + z = 2

3. 2x + 3y - z = 9

We can solve this system using various methods such as substitution, elimination, or matrices. Let's use the elimination method to solve this system:

1. Add equations (1) and (2):

(x + y - 2z) + (-x + 2y + z) = 5 + 2

x - x + y + 2y - 2z + z = 7

3y - z = 7   (Equation 4)

2. Multiply equation (2) by 2 and add it to equation (3):

2(-x + 2y + z) + (2x + 3y - z) = 2 × 2 + 9

-2x + 4y + 2z + 2x + 3y - z = 4 + 9

7y + z = 13  (Equation 5)

Now we have equations (4) and (5):

4. 3y - z = 7

5. 7y + z = 13

Let's solve this system by adding equations (4) and (5):

(3y - z) + (7y + z) = 7 + 13

3y + 7y = 20

10y = 20

y = 2

Now that we have found y = 2, we can substitute this value into either equation (4) or equation (5) to solve for z.

Let's use equation (4):

3(2) - z = 7

6 - z = 7

-z = 7 - 6

-z = 1

z = -1

Now that we have found y = 2 and z = -1, we can substitute these values into any of the original equations to solve for x.

Let's use equation (1):

x + 2 - 2(-1) = 5

x + 2 + 2 = 5

x + 4 = 5

x = 5 - 4

x = 1

Complete Question may be:

Solve the system of equations:

x + y - 2z = 5

-x + 2y + z = 2

2x + 3y - z = 9

perimeter = 2W+2L

Length = 4W

160 = 2W +2(4W)

160 = 10W

w = 160/10 = 16 yards

length = 16*4 = 64 yards

Check:

64*2 = 128, 16*2 = 32

128+32 = 160

Length = 64 yards, Width = 16 yards

Casey earned $780.25 in tips last week. This was calculated by first finding out her earnings from her hourly wage, and then subtracting this from her total earnings.

First, we need to calculate how much Casey earns from the hourly rate alone. Given that Casey's hourly rate is $4.55 per hour and she worked for 26 hours last week, we multiply the two to get her earnings from the hourly wage, which is $4.55 * 26 = $118.30.

Then we need to find out how much Casey made in tips. Casey's total earnings last week were $898.55. We subtract the earnings from her hourly rate from her total earnings to find out how much she made in tips. So, $898.55 - $118.30 = $780.25.

Therefore, Casey earned $780.25 in tips last week, which corresponds to option D.

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The point slope form of the line has the following form:

y – y1 = m (x – x1)

The slope m can be calculated using the formula:

m = (y2 – y1) / (x2 – x1)

m = (1 - -3) / (4 – 0) = 1

So the whole equation is:

y – -3 = 1 (x – 0)

y + 3 = x

Answer:

Step-by-step explanation:

(a) The attachment shows the graph of the parabola in blue. It also shows an inscribed rectangle in black.

(b) The upper right point of the rectangle is shown in the attachment as (x, y). The dimension y is the height of the rectangle. The x-dimension is half the width of the rectangle, which is symmetrical about the y-axis. Hence the width is 2x.

(c) As with any rectangle, the area is the product of length and width:

... A = (2x)(9 -x²) . . . . . the attachment shows a graph of this

... A = -2x³ +18x . . . . . expanded form suitable for differentiation

A suitable domain for A is where both x and A are non-negative: 0 ≤ x ≤ 3.

(d) The derivative of A with respect to x is ...

... A' = -6x² +18

This is defined everywhere, so the critical values will be where A' = 0.

... 0 = -6x² +18

... 3 = x² . . . . . . . divide by -6, add 3

... √3 = x . . . . . . . -√3 is also a solution, but is not in the domain of A

(e) The rectangle will have its largest area where x=√3. That area is ...

... A = 2x(9 -x²) = 2√3(9 -(√3)²) = 2√3(6)

... A = 12√3 . . . . square units . . . . ≈ 20.785 units²

The parabola y = 9 - x^2 is graphed, and a rectangle's vertices and area calculations are superimposed on it. The derivative of the area function gives critical values, determining the maximum area is 18 square units.

To solve this problem, we must utilize quadratic equations, Two-Dimensional (x-y) Graphing, and calculus.

(a) To begin, we need to draw a graph of the parabolic equation y = 9 - x^2. The vertex of the parabola is at the point (0,9) since the x-coordinate of the vertex is -b/2a in a general quadratic equation of the form y = ax^2 + bx + c, and here a = -1 and b = 0.

(b) For the rectangle, the upper right vertex is at point (x, y=(9 - x^2)), with base length 2x and height (9 - x^2).

(c) The area (A=base * height) of the rectangle is then given by A = 2x * (9 - x^2). The domain of A is specified by the values of x for which A is defined. Here A is defined for all x ∈ (-∞, ∞).

(d) The derivative of A with respect to x is A’ = 2(9 - 3x^2), which can be set equal to zero, the solution of which gives critical values x = ±√3.

(e) By the second derivative test or comparing values at the endpoints and critical points, you can determine that the largest area is 18 square units when x = √3.

Therefore,

b) The upper right vertex is at point (x, y=(9 - x^2)), with base length 2x and height (9 - x^2).

c) Here, A is defined for all x ∈ (-∞, ∞).

d) The solution of which gives critical values x = ±√3.

e) The largest area is 18 square units.

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The formula for finding the slope of a line is expressed as:

[tex]m = \frac{y_2-y_1}{x_2-x_1}[/tex]

Given the coordinate points (2, 1) and (2, 0). Get the slope of the line passing through the coordinate points

[tex]m=\frac{0-1}{2-2}\\m=\frac{-1}{0}\\m=\infty[/tex]

The equation of the line in point-slope form is [tex]y-y_0=m(x-x_0)[/tex]

Since the slope of a vertical line does not exist, hence the equation of the line cannot be written in point-slope form

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The perimeter of the rectangle with four times longer in length than its width and an area of 400 cm2 is 100 cm.

To solve this, let's assume the width of the rectangle is w centimeters. This would make the length 4w centimeters. Since the area of a rectangle is given by the product of its length and width, we have:

w x 4w = 400

Solving for w, we find that w2 = 100, so w = 10 cm.

Hence, the length is 4 x 10 = 40 cm.

The perimeter of a rectangle is given by the formula 2l + 2w, so substituting in our values:

Perimeter = 2(40) + 2(10) = 80 + 20 = 100 cm.

Therefore, the perimeter of the rectangle is 100 cm.

Answer:

The rate of change of the volume of the cylinder when the radius is 10 ft is

[tex]\frac{dV}{dt}=400\pi \:{\frac{ft^3}{min} }[/tex]

Step-by-step explanation:

This is a related rates problem. A related rates problem is a problem in which we know the rate of change of one of the quantities (the height of a grain) and want to find the rate of change of the other quantity (the volume of grain in the cylinder).

The volume of a cylinder is given by

[tex]V=\pi r^2 h[/tex]

V and h both vary with time so you can differentiate both sides with respect to time, t, to get

[tex]\frac{dV}{dt}=\pi r^2 \frac{dh}{dt}[/tex]

Now use the fact that [tex]\frac{dh}{dt} = 4 \:{\frac{ft}{min} }[/tex] and [tex]r = 10 \:ft[/tex]

[tex]\frac{dV}{dt}=\pi (10)^2 \cdot 4\\\\\frac{dV}{dt}=100\cdot \:4\pi \\\\\frac{dV}{dt}=400\pi[/tex]

A binary search in a set of 64 sorted items will, in the worst-case scenario, require 6 comparisons. This is calculated as log2(64) = 6. The binary search method works by repeatedly halving the search interval until it finds the desired item.

In order to find out how many comparisons are needed for a binary search in a set of 64 elements, we need to understand how a binary search works. It's an efficient algorithm that finds an item from a sorted list by repeatedly dividing the search interval in half. Every time it analyzes the middle element, it either discards the half that it is sure does not contain the desired item, or determines that the middle element is the desired item.

With a set of 64 elements, a binary search would perform log2(64) or 6 comparisons, as the base-2 logarithm of 64 is 6. Therefore, the maximum number of comparisons necessary using a binary search to locate a particular item in a set of 64 elements is 6 comparisons.

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In a binary search of a 64 element set, the worst-case scenario requires 6 comparisons. This is determined by the base 2 logarithm of the number of elements.

The number of comparisons in a binary search is governed by the logarithm base 2 of the number of elements in the set. With a set of 64 elements, the worst case scenario for the number of comparisons needed would be log2(64) which equals 6. This is because a binary search works by repeatedly dividing the searchable set in two until it finds the desired element, making it a very efficient search algorithm.

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The odds against selecting a person who commutes by bicycle from a town where 10% of people commute that way would be 9 to 1. This is based on the percentage of people who do not commute by bicycle versus those who do.

In statistical terms, the question is asking us to calculate the odds against selecting someone who commutes by bicycle. With 10% of people in this town commuting by bicycle, it means that 90% of them do not. Therefore, the odds against selecting someone who commutes by bicycle are 90 to 10 or 9 to 1.

This is calculated by dividing the number of unsuccessful outcomes (people not commuting by bicycle - 90%) by the number of successful outcomes (people commuting by bicycle - 10%). This gives us the odds against selecting someone who commutes by bicycle.

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The height of the air balloon is 63.4 yards.

Trigonometry is a branch of mathematics that uses variables to determine heights and distances. It is the study of the properties of right angled triangles and trigonometric functions and of their applications.

For the given situation,

The diagram below shows the situation described above.

The distance between Caleb and Emily = 100 yards

Let the distance from Caleb to the air balloon be 'x'

Let the distance from Emily to the air balloon be 'y'

Let height of the air balloon be 'h'

⇒ [tex]x+y=100[/tex]

⇒ [tex]x=100-y[/tex]

We know that, [tex]tan[/tex] θ = [tex]\frac{perpendicular }{base}[/tex]

Now consider the triangle with angle 45°,

[tex]tan 45[/tex]° = [tex]\frac{h}{x}[/tex]

We know that tan 45° = 1 and substitute x = 100-y,

⇒ [tex]1 = \frac{h}{100-y}[/tex]

⇒ [tex]h=100-y[/tex]

This is equation 1.

Now consider the triangle with angle 60°,

[tex]tan60[/tex]° = [tex]\frac{h}{y}[/tex]

We know that, tan 60° = √3

⇒ [tex]\sqrt{3} = \frac{h}{y}\\[/tex]

⇒ [tex]h=y\sqrt{3}[/tex]

This is the equation 2.

On equating equation 1 and 2,

⇒ [tex]100-y=y\sqrt{3}[/tex]

⇒ [tex]y+y\sqrt{3}=100[/tex]

⇒ [tex]y(\sqrt{3}+1 )=100[/tex]

⇒ [tex]y=\frac{100}{\sqrt{3}+1 }[/tex]

Thus height, [tex]h=y\sqrt{3}[/tex]

Substitute the value of y in h,

⇒ [tex]h=\sqrt{3}(\frac{100}{\sqrt{3} +1} )[/tex]

⇒ [tex]h=\frac{100\sqrt{3} }{\sqrt{3}+1 }[/tex]

⇒ [tex]h=\frac{173.205}{2.732}[/tex]

⇒ [tex]h=63.397[/tex]

⇒ [tex]h=63.4[/tex]

Hence we can conclude that the height of the air balloon is 63.4 yards.

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The total number of grandchildren of Kamiko and her 4 sisters is 72.

Kamiko and her 4 sisters each have 18 grandchildren. To calculate the total number of grandchildren, we will multiply the number of grandchildren per person by the total number of sisters.

Number of grandchildren per person: 18

Total number of sisters: 4

Therefore, the total number of grandchildren of Kamiko and her 4 sisters is 72.

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