Type the correct answer in the box. The value of an autographed baseball from 2017 is [tex]$300[/tex]. The value of the baseball exponentially increases by 5% each year after 2017. Write a one-variable inequality that could be used to solve for the number of years, [tex]$x$[/tex], it would take for the baseball to be worth at least [tex]$650[/tex]. Do not use any dollar signs in the inequality. Write the inequality in standard form.
Answer (2)
The problem states that the value of a baseball increases exponentially by 5% each year from an initial value of $300.
We express the value after x years as 300 ( 1.05 ) x .
We set up the inequality 300 ( 1.05 ) x ≥ 650 to find when the value is at least $650.
The final inequality is 300 ( 1.05 ) x ≥ 650 .
Explanation
Understanding the Problem We are given that the initial value of an autographed baseball in 2017 is 300. T h e v a l u eo f t h e ba se ba ll in cre a sese x p o n e n t ia ll y b y 5 x$, it takes for the baseball to be worth at least $650.
Setting up the Inequality Let V ( x ) be the value of the baseball after x years. The exponential growth can be modeled by the equation V ( x ) = 300 ( 1 + 0.05 ) x = 300 ( 1.05 ) x . We want to find the number of years x such that V ( x ) ≥ 650 . Therefore, we need to solve the inequality 300 ( 1.05 ) x ≥ 650 .
Simplifying the Inequality To solve the inequality 300 ( 1.05 ) x ≥ 650 , we can divide both sides by 300: ( 1.05 ) x ≥ 300 650 = 6 13
Final Answer So, the inequality that could be used to solve for the number of years, x , it would take for the baseball to be worth at least 650 i s : 300 ( 1.05 ) x ≥ 650 $
Examples
Exponential growth is a mathematical transformation that increases without bound. A real-world example is compound interest. Suppose you invest $1000 in an account that pays 5% interest compounded annually. After one year, you'll have $1000 * (1 + 0.05) = $1050. After two years, you'll have $1000 * (1.05)^2 = $1102.50. This exponential growth continues, illustrating how investments can grow significantly over time.
The inequality that can be used to find how many years it takes for an autographed baseball valued at 300 dollars in 2017 to be worth at least 650 dollars, increasing at 5% annually, is 300(1.05)^x ≥ 650. This represents the relationship between the initial value, growth rate, and desired final value. Dividing both sides by 300 gives an alternative form: (1.05)^x ≥ 13/6.
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