One of the rides in the park sends the rider down a steep slide and over a series of hills that
gradually decrease in height until the ride ends. Using the table below, create both a recursive and
an explicit geometric equation for the height of a given hill. Use your recursive equation to answer
the following question: if you increased the drop height to 400, what would be the height of the
fourth hill?
0
1
2
320
80
20
5
1.25
To create a recursive equation, we can observe that the height of each hill is one-fourth the height of the previous hill:
Recursive equation: H(n) = H(n-1) * (1/4)
Using this equation, we can find the height of the fourth hill:
H(4) = H(3) * (1/4)
= 20 * (1/4)
= 5
Therefore, the height of the fourth hill would be 5 units.
To create an explicit geometric equation, we can observe that each hill's height is equal to the initial drop height (h0) multiplied by the common ratio (r) raised to the power of the hill number (n):
Explicit geometric equation: H(n) = h0 * r^(n-1)
In this case, the initial drop height is 320, and the common ratio is 1/4:
H(n) = 320 * (1/4)^(n-1)
To find the height of the fourth hill using this equation:
H(4) = 320 * (1/4)^(4-1)
= 320 * (1/4)^3
= 320 * (1/64)
= 5
Again, the height of the fourth hill is 5 units.
To create a recursive equation for the height of each hill, we can observe that each subsequent hill is 1/4th the height of the previous hill.
Let's define the recursive equation as h(n), where n represents the hill.
Recursive equation: h(n) = h(n-1) / 4, with the initial condition of h(0) = 0.
Using this recursive equation, we can calculate the height of the fourth hill (n = 4) by repeatedly applying the formula:
h(4) = h(3) / 4
h(3) = h(2) / 4
h(2) = h(1) / 4
h(1) = h(0) / 4
Using the given table, we can substitute the values into the recursive equation:
h(1) = 320
h(2) = 80
h(3) = 20
Now, let's substitute the values into the recursive formula:
h(4) = 20 / 4
= 5
Therefore, the height of the fourth hill would be 5.
Alternatively, we can also determine the explicit geometric equation for the height of each hill.
Using the given table, we can observe that each subsequent hill is 1/4th the height of the previous hill. This relationship can be expressed as:
Explicit geometric equation: h(n) = h(0) * (1/4)^n
Let's calculate the height of the fourth hill (n = 4) using this explicit geometric equation:
h(4) = 0 * (1/4)^4
= 0 * (1/256)
= 0
Therefore, using the explicit geometric equation, the height of the fourth hill would be 0.
To create a recursive equation for the height of a given hill, we need to identify the pattern in the table and determine how to calculate the height of a specific hill based on the heights of previous hills.
Looking at the table, we can see that each hill's height is formed by multiplying the previous hill's height by a constant, which is 0.25 in this case. However, for the first hill, the height is given as 320. This gives us the base case for our recursive equation.
Recursive Equation:
- If n = 1, height of the first hill (h1) = 320 (base case)
- If n > 1, height of the nth hill (hn) = (height of the previous hill) * 0.25
Explicit Geometric Equation:
The explicit equation for the geometric sequence represented by the heights of the hills can be derived using the formula for geometric sequence:
hn = a * r^(n-1)
where a is the first term, r is the common ratio, and n is the term number.
In this case, using the value given in the table, the explicit equation would be:
hn = 320 * 0.25^(n-1)
To find the height of the fourth hill using the recursive equation, we start with the base case and apply the recursive rule three times:
h1 = 320
h2 = h1 * 0.25 = 320 * 0.25 = 80
h3 = h2 * 0.25 = 80 * 0.25 = 20
h4 = h3 * 0.25 = 20 * 0.25 = 5
Therefore, the height of the fourth hill, using the recursive equation, would be 5.
To find the height of the fourth hill using the explicit equation, we substitute n = 4 into the equation:
h4 = 320 * 0.25^(4-1)
= 320 * 0.25^3
= 320 * 0.015625
= 5
Again, we get the same result of 5 for the height of the fourth hill using the explicit equation.