1. The distance, d(t), fallen by a free-falling body is represented by d(t)+vt-0.5gt^2, where v is initial velocity and g is acceleration due to gravity. The acceleration due to gravity is 9.8 m/s^2. If a rock is thrown downward with an initial velocity of 4 m/s from the edge of the North Rim of the Grand Canyon, which is 1750 meters deep, how long will it take the rock to reach the bottom of the Grand Canyon?
2.The length of a rectangular garden is 6ft. more than its width. A path 3ft. wide surrounds the outside of the garden. The total area of the path is 288 square feet. Find the dimensions of the garden.
3. Ron and Mary Beth are hired to stock shelves at a supermarket. It takes them 4 hours working together to complete the job. It would take Ron 6 more hours than Mary Beth to do the job alone. If one of them were absent how long would the other one need to complete the job alone?
4.Determine if a rectangular piece of paper with a perimeter of 36 inches can have an area of 81in^2. State the dimensions if it does exist.
#1 -4t-4.9t^2 = -1750
#2 (w+2*3)((w+6)+2*3)-(w)(w+6) = 288
#3 If Mary Beth takes x hours,
1/x + 1/(x+6) = 1/4
#4 clearly a 9x9 square fits the bill
Since a square has maximum area for a given perimeter, no other rectangle can do the job.
Thank you so much for all the help!
1. To find the time it takes for the rock to reach the bottom of the Grand Canyon, we need to set up an equation using the given information and solve for t.
The equation for the distance fallen is d(t) = vt - 0.5gt^2, where d(t) is the distance fallen at time t, v is the initial velocity, and g is the acceleration due to gravity (9.8 m/s^2).
In this case, the initial velocity is 4 m/s, and the distance fallen is 1750 meters.
So we have the equation: 1750 = (4)t - 0.5(9.8)t^2.
To solve for t, we can rearrange the equation and set it equal to zero:
0.5(9.8)t^2 - (4)t + 1750 = 0.
Using the quadratic formula, t = (-b ± √(b^2 - 4ac)) / (2a), where a, b, and c are the coefficients of the equation, we can find the two possible values of t.
Plugging in the values, we have a = 0.5(9.8) = 4.9, b = -4, and c = 1750.
t = [4 ± √((-4)^2 - 4(4.9)(1750))] / (2(4.9)).
Solving this equation will give us the two possible values of t. We can discard any negative values as they don't make sense in this context.
2. To find the dimensions of the rectangular garden, we can set up an equation using the given information.
Let's assume the width of the garden is x ft.
According to the problem, the length of the garden is 6 ft more than its width, so the length would be (x + 6) ft.
The total area of the path surrounding the garden is 288 square feet. The width of the path is 3 ft, so the measurements of the outer rectangle would be (x + 2(3)) and ((x + 6) + 2(3)) ft.
We can calculate the area of the outer rectangle by multiplying its length by its width:
Area of the outer rectangle = (x + 6 + 2(3))(x + 2(3)).
= ((x + 12)(x + 6)).
Then, we subtract the area of the actual garden from the area of the outer rectangle to get the area of the path:
Area of the path = (x + 12)(x + 6) - (x)(x + 6).
We know the area of the path is 288 square feet, so we have:
288 = (x + 12)(x + 6) - (x)(x + 6).
Simplifying and solving this equation will give us the dimensions of the garden.
3. To find out how long it would take each person to complete the job alone, we can set up an equation using the given information.
Let's assume Ron can complete the job alone in x hours. According to the problem, it takes Ron 6 more hours than Mary Beth to do the job alone, so Mary Beth would take (x - 6) hours.
Working together, they can complete the job in 4 hours, so their combined work rate is 1/4 of the job per hour.
To find how long each person would take to complete the job alone, we can set up the following equation:
1/x + 1/(x - 6) = 1/4.
To solve this equation, we can find a common denominator and simplify.
Once we solve for x, we can find the time it would take for each person to complete the job alone by substituting the value of x into the respective equations (x hours for Ron, and (x - 6) hours for Mary Beth).
4. To determine if a rectangular piece of paper with a perimeter of 36 inches can have an area of 81in^2, we can use the formulas for perimeter and area of a rectangle.
The perimeter of a rectangle is given by the formula: Perimeter = 2(length + width).
In this case, we have a perimeter of 36 inches, so we can set up the equation:
36 = 2(length + width).
Additionally, the area of a rectangle is given by the formula: Area = length * width.
In this case, we have an area of 81 square inches, so we can set up the equation:
81 = length * width.
We can solve these two equations simultaneously to determine if there is a solution. If the length and width satisfy both equations, then the dimensions of the rectangle can be stated.