The probability P, as a percent, that a certain computer keyboard in a public library will last more than t years can be approximated by
P = 100e^−1.2t.
After how many years will the probability of keyboard failure be 76%? Round to the nearest tenth.
.76 = e^-1.2 t
ln .76 = -1.2 t
just like your other problem
Well, in order to find the number of years when the probability of keyboard failure is 76%, we can use a little math. We'll rearrange the equation to solve for t:
P = 100e^(-1.2t)
76 = 100e^(-1.2t)
Now, let's do some equation acrobatics here. First, divide both sides of the equation by 100:
0.76 = e^(-1.2t)
Next, take the natural logarithm of both sides:
ln(0.76) = -1.2t
Now, divide both sides by -1.2:
t = ln(0.76) / -1.2
And when you calculate it out, you'll find that t is approximately 1.17 years.
So after about 1.2 years, there's a 76% chance your keyboard will fail. But hey, at least you have some time to enjoy the clackety-clack of those keys!
To find the number of years after which the probability of keyboard failure will be 76%, we need to solve the equation:
76 = 100e^(-1.2t)
To isolate the exponential term, we divide both sides of the equation by 100:
76/100 = e^(-1.2t)
0.76 = e^(-1.2t)
To get rid of the exponential term, we can take the natural logarithm (ln) of both sides:
ln(0.76) = ln(e^(-1.2t))
Using the property of logarithms that ln(e^x) = x, we simplify the equation:
ln(0.76) = -1.2t
Now, we can solve for t by dividing both sides by -1.2:
t = ln(0.76) / -1.2
Using a calculator, we find:
t ≈ 3.4
Therefore, after approximately 3.4 years, the probability of keyboard failure will be 76%.
To find out after how many years the probability of keyboard failure will be 76%, we need to solve the equation:
P = 100e^(-1.2t)
where P represents the probability (in percentage) and t represents the number of years.
In this case, we know that P = 76%. Let's substitute this value into the equation:
76 = 100e^(-1.2t)
To solve for t, we need to isolate the exponential term on one side of the equation. Divide both sides of the equation by 100:
76/100 = e^(-1.2t)
Simplify:
0.76 = e^(-1.2t)
Next, we need to take the natural logarithm of both sides of the equation to eliminate the exponential term:
ln(0.76) = ln(e^(-1.2t))
Applying the logarithm rules, we can bring the exponent down as a multiplier:
ln(0.76) = -1.2t ln(e)
Since ln(e) equals 1, the equation simplifies to:
ln(0.76) = -1.2t
Now, divide both sides of the equation by -1.2:
t = ln(0.76) / -1.2
Using a calculator or logarithmic tables, calculate ln(0.76) and divide the result by -1.2. The result will give you the number of years after which the probability of keyboard failure will be 76%. Round the answer to the nearest tenth.