1. Which phrase describes an unknown or changeable quantity
A. 4 quarts in a gallon
B. 3 feet 7 inchits
C. 2 o’clock in the afternoon
D. The height of the building times (1)/(2)
2. the loall gym currently has 100 members. the membership entrollment is increasing at a rate of 8% per year. write an exponential equation for this situation.
The Correct Answer and Explanation is :
1. Which phrase describes an unknown or changeable quantity?
The correct answer is D. The height of the building times (1)/(2).
This option involves a variable that could change depending on the height of the building. The height of the building is an unknown quantity that can vary, and multiplying it by 1/2 indicates a relationship that can change as the height changes. Therefore, this is the correct choice to describe a changeable quantity.
The other options are specific, fixed quantities:
- A. 4 quarts in a gallon is a fixed conversion factor.
- B. 3 feet 7 inches is a fixed measurement of length.
- C. 2 o’clock in the afternoon is a specific time and does not vary in the context given.
2. Exponential Equation for Gym Membership Enrollment
The gym has 100 members initially, and membership is increasing at a rate of 8% per year. To write an exponential equation that models this situation, we use the general form of an exponential growth equation:
[
P(t) = P_0 \times (1 + r)^t
]
Where:
- (P(t)) is the population (or number of gym members) at time (t),
- (P_0) is the initial population (or number of members at the start),
- (r) is the growth rate (as a decimal),
- (t) is the time in years.
Given values:
- Initial members ((P_0)) = 100
- Growth rate ((r)) = 8% = 0.08 (as a decimal)
- Time ((t)) is the number of years.
So, the exponential equation for this situation becomes:
[
P(t) = 100 \times (1 + 0.08)^t
]
[
P(t) = 100 \times (1.08)^t
]
Explanation:
The formula represents exponential growth, where the number of gym members increases by a constant percentage every year. Here’s a breakdown of how it works:
- Initial Population (100): The term (P_0 = 100) reflects that there are 100 members to start with.
- Growth Rate (8%): The growth rate is 8%, which is expressed as a decimal (0.08). Each year, the gym’s membership increases by 8% of the current membership.
- Time ((t)): The variable (t) represents the number of years that have passed. As time increases, the number of gym members will increase exponentially because of the compounding growth rate.
At the end of the first year, for example, the number of members would be:
[
P(1) = 100 \times 1.08 = 108 \text{ members}
]
At the end of the second year, the number would increase again by 8%, so:
[
P(2) = 100 \times 1.08^2 \approx 116.64 \text{ members}
]
This equation is useful for predicting future membership growth based on the given rate of increase.