A car is sold for $22,000. After one year, the value of the car is $16,500. Write an exponential function y to determine the value of the car after x years if the rate of decrease is the same each year.
since 16500/22000 = 0.75,
y = 22000 * 0.75^x
Sure, let's find the funny side of exponential decay. I hope it will drive you to laughter!
Now, let's find the equation for the value of the car. We know that the rate of decrease is the same each year, so we can use the formula for exponential decay, which is y = a * (1 - r)^x.
In this case, the initial value is $22,000 (a), and after one year the value is $16,500 (y). So our equation becomes:
$16,500 = $22,000 * (1 - r)^1.
Now, I don't want to "car"ry on too much, but let's solve this equation step by step! Divide both sides by $22,000:
$16,500 / $22,000 = (1 - r)^1.
Simplify this a bit, and we are left with:
0.75 = (1 - r)^1.
To get rid of the pesky exponent, we'll bring it back down to earth by taking the root of both sides:
√0.75 = √(1 - r)^1.
Surprise! Taking the square root of 0.75 gives us approximately 0.866! So, we get:
0.866 = 1 - r.
Now, our rapidly shrinking car calls for some comedy magic! Let's move the -r to the other side:
r = 1 - 0.866.
Calculating this slight "wheelie" of a subtraction, r is equal to approximately 0.134.
And finally, we can substitute this value back into our initial equation to get the full formula for our car's value over time:
y = $22,000 * (1 - 0.134)^x.
There you have it! The exponential function that will make your car's value "depreciate" in style.
To write an exponential function to determine the value of the car after x years, we need to find the rate of decrease per year.
Let's assume the initial value of the car is A dollars and it decreases by r% per year.
Using the given information:
After one year, the value of the car is $16,500, which is 100% - r% of the initial value.
So, A - (r/100) * A = 16,500
Simplifying the equation:
(100 - r)A / 100 = 16,500
Now, we know that the initial value of the car is $22,000:
(100 - r) * 22,000 / 100 = 16,500
Next, we can solve for r:
(100 - r) * 22,000 = 16,500 * 100
Simplifying further:
2,200,000 - 22,000r = 1,650,000
Now, isolate the variable r:
-22,000r = 1,650,000 - 2,200,000
-22,000r = -550,000
r = -550,000 / -22,000
r = 25
Therefore, the rate of decrease is 25% per year.
Now, let's write the exponential function y to determine the value of the car after x years:
y = A * (1 - r/100)^x
Plugging in the value for A ($22,000) and r (25% or 0.25) into the function, we get:
y = 22,000 * (1 - 0.25)^x
Thus, the exponential function to determine the value of the car after x years is:
y = 22,000 * 0.75^x
To determine an exponential function to represent the value of the car over time, we need to use the given information about the rate of decrease.
We know that the initial value of the car is $22,000, and after one year, the value is $16,500. This means that the value decreased by $22,000 - $16,500 = $5,500 in one year.
Let's denote the value of the car after x years as y. We can write the exponential function as follows:
y = ab^x
In this case, we know that the initial value is $22,000, so a = 22000. We also know that the car's value decreased by $5,500 each year, so we can calculate the common ratio b as follows:
b = (Value after one year) / (Initial value)
b = 16500 / 22000
b = 0.75
Therefore, the exponential function to determine the value of the car after x years, considering the same rate of decrease each year, is:
y = 22000 * (0.75)^x
This function will give you the value of the car after x number of years.