As a town gets smaller, the population of its high school decreases by 8% each year. The senior class has 320 students now. In how many years will it have about 100 students? Write an equation. Then solve the equation without graphing.

Question

As a town gets​ smaller, the population of its high school decreases by 8​% each year. The senior class has 320  students now. In how many years will it have about 100 ​students? Write an equation. Then solve the equation without graphing.

GPT 3.5 · Answer

Let's assume that the number of years it will take for the senior class to have about 100 students is x.

Since the population decreases by 8% each year, the population after x years can be expressed as: 320 * (1 - 0.08)^x.

Setting this expression equal to 100, we can write the equation: 320 * (1 - 0.08)^x = 100.

Now, let's solve this equation without graphing:

Divide both sides of the equation by 320: (1 - 0.08)^x = 100/320.

Simplify 100/320: (1 - 0.08)^x = 0.3125.

Take the logarithm base 10 of both sides of the equation: log((1 - 0.08)^x) = log(0.3125).

Use the logarithmic power rule to bring the exponent down: x * log(1 - 0.08) = log(0.3125).

Divide both sides of the equation by log(1 - 0.08): x = log(0.3125)/log(1 - 0.08).

Using a calculator, evaluate: x ≈ -1.933.

Since the number of years cannot be negative, we'll take the absolute value of x: x ≈ 1.933.

Therefore, it will take approximately 1.933 years for the senior class to have about 100 students.

As a town gets​ smaller, the population of its high school decreases by 8​% each year. The senior class has 320 students now. In how many years will it have about 100 ​students? Write an equation. Then solve the equation without graphing.

As a town gets smaller, the population of its high school decreases by 8% each year. The senior class has 320 students now. In how many years will it have about 100 students? Write an equation. Then solve the equation without graphing.