A motorcycle bought for \( \$ 10,000 \) depreciates continuously at \( 9 \% \) per annum. What is its value after 7 years? Round the answer to nearest dollar. Select the correct answeb.below:

To find the value of the motorcycle after 7 years, you can use the formula for continuous depreciation, which is given by the equation: \[ V = P e^{-rt} \] Where: - \( V \) is the future value, - \( P \) is the initial value (or purchase price), - \( r \) is the rate of depreciation (as a decimal), - \( t \) is the time in years, - \( e \) is the base of the natural logarithm (approximately equal to 2.71828). In this case: - \( P = 10,000 \) - \( r = 0.09 \) - \( t = 7 \) Plugging in the values: \[ V = 10,000 \times e^{-0.09 \times 7} \] \[ V \approx 10,000 \times e^{-0.63} \approx 10,000 \times 0.5308 \approx 5,308 \] So, the value of the motorcycle after 7 years would be approximately **$5,308** when rounded to the nearest dollar. Now, let’s think about how such depreciation continues through time in fun, engaging ways. You know, motorcycles are awesome machines, but just like any vehicle, they can lose their value over time based on various factors! Continuous depreciation is one interesting concept that applies to many things, especially tech gadgets and cars! So yeah, hanging onto your favorite ride might just make you a real motorcycle enthusiast! But when you're figuring out the depreciation, remember the importance of the exponential factor! It's easy to mix it with simple depreciation, leading to underestimating the actual loss. Double-checking your numbers is key! Whether you bring a calculator or old-school pencil and paper, keep that meticulous spirit alive—your future self will thank you!