Laura drove to the mountains last weekend. there was heavy traffic on the way there, and the trip took 8 hours. when laura drove home, there was no traffic and the trip only took 6 hours. if her average rate was 16 miles per hour faster on the trip home, how far away does laura live from the mountains

a) The radius of the navy pier wheel is 70 feet.

b) The radius of the first ferris wheel is 125 feet.

c) The speed of the wheel is 87.2 ft/sec

Given data:

The circumference of circle = 2πr

The area of the circle = πr²

a)

The circumference = 2πr

where π = 3.142

r = radius

Therefore, 2πr = 439.6

(2 * 3.142 * r) = 439.6

6.284r = 439.6

On simplifying the equation:

Radius = 439.6/6.284

Radius = 69.9

≈ 70 feet

b)

The Navy Pier Ferris Wheel in Chicago has a circumference that is ​56% of the circumference of the first Ferris wheel.

So, 56% of f = 439.6

0.56f = 439.6

f = 439.6/0.56

f = 785

The circumference of the first wheel is 785 feet.

Now, 2πr = 785

r = 785/2π

r = 785/(2×3.142)

r = 785/6.284

r = 124.9

≈ 125ft

c)

The first ferris wheel took nine minutes to make a complete revolution.

So, the speed of wheel is determined as:

s = 785/9

s = 87.2 ft/sec

To learn more about circle, refer:

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The complete question is attached below:

If the navy pier ferris wheel in chicago has a circumference that is 56% of the circunference of the first ferris wheel built in 1893.

a. What is the radius of the navy pier wheel?

b. what was the radius of the first ferris wheel?

c. The first ferri wheel took nine minutes to make a complete revolution. how fast was the wheel moving?

C=439.6 ft.

The tip of the minute hand moves approximately (8/3)π inches in 20 minutes, using the circumference of the circular path it follows and the proportion of the path covered in that time.

The student is asking about the distance traveled by the tip of the minute hand of a clock over a 20-minute period. To find the distance, we'll consider the movement of the minute hand as part of a circular path, which is described by the arc length of the circle segment.

The length of the minute hand, which is 4 inches, acts as the radius (r) of the circle. The minute hand completes one full rotation around the clock face in 60 minutes, so in 20 minutes, it covers one-third of a full rotation. We can find the arc length of the minute hand's path using the formula for the circumference of a circle (C = 2πr), multiplied by the fraction of the rotation:

Circumference: C = 2π(4 inches) = 8π inches

Arc Length for 20 minutes: (1/3) × 8π inches = (8/3)π inches

Therefore, the tip of the minute hand moves approximately (8/3)π inches in 20 minutes.

Indication "12/10, n/30" (or "12/10 net 30") on an account represents a cash (sales) discount provided by the seller to the buyer for swift payment.

The term 12/10, n/30 is a classic credit term and means the following:

The discount date begins at August 9 and the last day would be August 19.

The '12/10, n/30' term means a 12% discount is given if payment is made within 10 days of the invoice date, and the entire invoice must be paid within 30 days. As such, the discount date is August 19.

The sale terms 12/10, and n/30 in the question denote a sale transaction where the seller is offering the buyer a 12% discount if they pay the invoice within 10 days of the invoice date. The 'n/30' part means that the rest of the invoice amount needs to be paid within 30 days. So, if the manager of the new business opts to take the cash discount, the discount date falls on August 19 (the invoice date - August 9 plus 10 days).

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[tex]\(x^2 - x - 6\)[/tex] can be expressed as the product of [tex]\((x - 3)\) and \((x - 2)\)[/tex]. The long division image is given below.

Long division for polynomials is similar to long division for numbers. We'll divide the quadratic polynomial [tex]\(f(x) = x^2 - x - 6\)[/tex] by the binomial [tex]\((x - 3)\)[/tex].

Here's a step-by-step breakdown of the process:

1. Start by dividing the highest-degree term of the dividend (in this case, [tex]\(x^2\)[/tex]) by the highest-degree term of the divisor (in this case, [tex]\(x\))[/tex]. This gives us [tex]\(x\)[/tex].

2. Multiply the divisor [tex]\((x - 3)\)[/tex] by the result from step 1, which is [tex]\(x\),[/tex] and write the result below the dividend, aligning like terms. So, [tex]\(x \cdot (x - 3) = x^2 - 3x\)[/tex].

3. Subtract the result from step [tex]2 (\(x^2 - 3x\))[/tex] from the dividend ([tex]\(x^2 - x - 6\)[/tex]). This gives us [tex]\(x^2 - x - 6 - (x^2 - 3x)\)[/tex], which simplifies to [tex]\(-x + 3x - 6\),[/tex] resulting in [tex]\(2x - 6\)[/tex].

4. Now, we repeat the process with the new expression [tex]\(2x - 6\)[/tex]. We divide the highest-degree term ([tex]\(2x\)[/tex]) by the highest-degree term of the divisor [tex](\(x\)),[/tex] which gives us [tex]\(2\).[/tex]

5. Multiply the divisor [tex]\((x - 3)\)[/tex] by the result from step 4, which is [tex]\(2\)[/tex], and write the result below the previous result: [tex]\(2 \cdot (x - 3) = 2x - 6\)[/tex].

6. Subtract the result from step [tex]5 (\(2x - 6\))[/tex] from the previous expression [tex](\(2x - 6\))[/tex], which results in [tex]\(0\)[/tex].

Since we have a remainder of [tex]\(0\),[/tex] this means that [tex]\(x^2 - x - 6\)[/tex] is evenly divisible by [tex]\((x - 3)\)[/tex], and the quotient is [tex]\(x - 2\).[/tex] The final result is:

[tex]\[ \frac{x^2 - x - 6}{x - 3} = x - 2 \][/tex]

This tells us that [tex]\(x^2 - x - 6\)[/tex] can be expressed as the product of [tex]\((x - 3)\) and \((x - 2)\)[/tex].

The answer to the statement is:

                     TRUE

In order for a segment to form a triangle.

The sum of the lengths of any two sides of a triangle must be greater than the length of the remaining third side of the given triangle.

Here we have three segments of length:

7 units, 9  units and 12 units.

Also, for any two pair:

7 and 9 we have:

7+9=16> 12

for 9 and 12 we have:

9+12=21>7

and for 7 and 12 we have:

7+12=19>9

Hence, the segments could form a triangle.

width = w

 length = w+8

perimieter = 256 feet

formla for perimeter is p=2L+2w

so you have: 256 =2(w+8) +2w =

256 = 2w+16 +2w

256= 4w+16

240=4w

w=240/4 = 60

 width is 60 feet

length = 60+8 = 68 feet

Answer:

x - 2y = -1

this is correct

The best approximation for the area of the smaller trapezoid is 39 [tex]ft^2[/tex].

Since the trapezoids are similar, the ratio of their corresponding side lengths is constant. We know the base of the larger trapezoid is 30 ft and the smaller one is 24 ft. Therefore, the ratio of their bases is:

ratio of bases = [tex]\frac{30 ft}{24 ft}[/tex] = [tex]\frac{5}{4}[/tex]

The area of a trapezoid is given by:

Area = [tex]\frac{1}{2}[/tex] * base * height

For similar shapes, the ratio of their areas is proportional to the square of the ratio of their corresponding side lengths. In this case, the ratio of areas will be:

Ratio of areas = [tex](Ratio \ {of \ { bases)^2[/tex] = [tex]({\frac{5}{4} })^2[/tex] = [tex]\frac{25}{16}[/tex]

We know the area of the larger trapezoid is 61 [tex]ft^2[/tex]. Applying the ratio of areas:

Area of smaller trapezoid = (61 [tex]ft^2[/tex]) * [tex]\frac{16}{25}[/tex]

Area of smaller trapezoid ≈ 38.91 [tex]ft^2[/tex] ≈ 39[tex]ft^2[/tex]

Answer:

[tex]\frac{2}{5}[/tex]-pound of sand

Step-by-step explanation:

We have been given that Amber has [tex]\frac{4}{5}[/tex]-pound bag of colored sand. She uses 1/2 of the bag for the project.

We need to find 1/2 'of' the 4/5 as:

[tex]\frac{4}{5}\times \frac{1}{2}[/tex]

[tex]\frac{4\times1}{5\times2}[/tex]

[tex]\frac{2\times1}{5\times1}[/tex]

[tex]\frac{2}{5}[/tex]

Therefore, Amber used [tex]\frac{2}{5}[/tex]-pound of sand for the project.

Final answer:

The three even integers are 62, 64, and 66.

Explanation:

The sum of the first and third of three consecutive even integers is 128. To find the three even integers, we start by letting the first integer be x. Because the integers are consecutive even numbers, the second integer will be x + 2 and the third will be x + 4. The sum of the first and third integers gives us the equation:

x + (x + 4) = 128

Solving this equation:

Combine like terms: 2x + 4 = 128.

Subtract 4 from both sides: 2x = 124.

Divide both sides by 2: x = 62.

So, the first even integer is 62. Consequently, the second is 62 + 2 = 64, and the third is 62 + 4 = 66. Therefore, the three consecutive even integers are 62, 64, and 66.

Lets do the first one(7). Fist change 10x -2y = 3,which is in standard form to point slope form so you can see the slope.

10x - 2y = 3

-2y = -10x + 3

-2y/-2 = -10x / -2 + 3/-2

y = 5x - 3/2

The slope is the m in y = mx + b so our slope is 5x, well it wants you to find the equation of a line that passes through (0,1) that is parallel to the equation this (Parallel) mean has the same slope.The slope is 5. Now that we know the slope is 5 we can find the equation by using the point slope form of an equation of a line.

point slope form =  y – y1 = m(x – x1)

Remember m = slope = 5. Now plugin our x and y cord  from the point (0,1) and our slope of 5

y - y1 = m(x - x1)

y - 1 = 5(x - 0)

y - 1 = 5x - 0

y - 1+1 = 5x + 1

y= 5x + 1

So the point (0,1) passes through y = 5x + 1

Number 8 is already in slope intercept form so you don't have to do anything to find the slope. You can see that the slope is 7 so just plug your point into y - y1 = m(x - x1) and then put it into slope intercept form.

8) 

Point = (5,8)

Slope = 7

y - y1 = m(x - x1) 

y - 8 = 7(x - 5) 

y =  7x - 27

So for number 8 the point (5,8) passes through y = 7x - 27

(9) For nine they want an equation that is perpendicular, which means to flip the slope. For instance, if you have a slope of 2/5 the perpendicular slope would be 5/2. For 9 we see that the slope is 2/5 so its perpendicular slope would be 5/2. Now just plug your point into the point slope form equation.

Point = (-4, 6)

Slope = 2/5 

perpendicular slope = 5/2

y - 6 = 5/2(x - (-4))

y = 5/2 x + 16

So point (-4, 6) passes through y = 5/2 x + 16

(10) Point (4,1) passes through y = 7/5 x - 23/5

Final answer:

To find vertical asymptotes of a function, factor the denominator, set each factor equal to zero, and solve for x.

Explanation:

To find vertical asymptotes of a function, we need to determine the values of x that make the function approach infinity or negative infinity. Vertical asymptotes occur when the function approaches these values but never reaches them.

To find the vertical asymptotes, follow these steps:

Factor the denominator of the function.

Set each factor equal to zero and solve for x.

The values of x obtained in step 2 are the vertical asymptotes.

For example, let's find the vertical asymptotes of the function f(x) = 1/x. The denominator is x, which is already factored. Setting x = 0, we find that x = 0 is the vertical asymptote.