A boat is tied to a pier with a 25-foot rope so that the distance between the top of the pier and the top of the boat deck is 7 feet. If the water level s down 8 feet with the low tide , how much closer ( in feet) is the boat to the base of the pier? Assume the rope is always taut? Shows your work
high tide:
x^2 + 7^2 = 25^2
x^2 = 576
x = 24
low tide:
x^2 + 15^2 = 25^
x^2 = 400
x = 20
The distance change is 4 ft
To solve this problem, we need to first find the original distance between the boat deck and the base of the pier using the Pythagorean Theorem. Let's denote this distance as "x".
According to the problem, the distance between the top of the pier and the top of the boat deck is 7 feet. And we know that the rope tied to the boat is 25 feet long. So we can set up the following equation:
x^2 + 7^2 = 25^2
Now, solve for x:
x^2 + 49 = 625
x^2 = 625 - 49
x^2 = 576
x = √576
x = 24 feet
So, initially, the boat is 24 feet away from the base of the pier.
Now, with the low tide, the water level goes down by 8 feet. This means the boat will sink 8 feet. Therefore, the new distance between the boat deck and the base of the pier can be calculated as:
x_new = x - 8
x_new = 24 - 8
x_new = 16 feet
Therefore, after the low tide, the boat is 16 feet away from the base of the pier. So it is now 8 feet closer to the base of the pier.
To find out how much closer the boat is to the base of the pier, we need to determine the new distance between the top of the pier and the top of the boat deck after the water level goes down.
Let's break down the problem:
1. Initially, the distance between the top of the pier and the top of the boat deck is 7 feet.
2. When the water level goes down 8 feet with low tide, the boat will be lower in the water. This means that the distance between the top of the pier and the top of the boat deck will decrease by 8 feet.
Now, let's calculate the new distance:
Initial distance: 7 feet
Decrease due to low tide: 8 feet
New distance = Initial distance - Decrease due to low tide
New distance = 7 feet - 8 feet
To subtract 8 from 7, we have to move 8 units in the opposite direction. Since it goes negative, we can write it as:
New distance = 7 feet + (-8 feet)
Now, we can perform the addition:
New distance = -1 foot
Therefore, the boat is 1 foot closer to the base of the pier after the water level goes down 8 feet with low tide.