4) The distance needed to bring a truck to a stop varies as the square of its speed. If a truck travels at a speed of 55 mph it needs 79 feet to stop. What is the stopping distance of the truck travelling at 70 mph?
5) A principal of $7000 is invested in an account paying an annual rate of 5%. Find the amount in the account after 5 years if the account is compounded semiannually, quarterly, and monthly.
d = (70/55)^2 * 79 = 128 Ft.
5. P = Po(1+r)^n.
Sem1-annually:
Po = $7,000.
r = (5%/2)/100% = 0.025 = Semi-annual %
rate expressed as a decimal.
n = 2Comp./yr. * 5yrs. = 10 Compounding
periods.
P = 7000 * (1.025)^10 = $8960.59.
Quarterly:
r = (5%/4)/100% = 0.0125 = Quarterly % rate expressed as a decimal.
n = 4comp./yr. * 5yrs. = 20 Compounding
periods.
P = ?.
Monthly:
r = (5%/12)/100% = 0.0042 = Monthly % rate expressed as a decimal.
n = 12comp./yr. * 5yrs. = 60 Compounding
periods.
P = ?.
4) To solve this problem, we can use the relationship between the speed (v) and the stopping distance (d) of the truck, which states that the stopping distance varies as the square of the speed.
We are given that when the truck is traveling at 55 mph, it needs 79 feet to stop. Let's denote this as (v1, d1), where v1 = 55 mph and d1 = 79 feet.
Now, we need to find the stopping distance of the truck when it is traveling at 70 mph. Let's denote this as (v2, d2), where v2 = 70 mph and we need to find d2.
Since the stopping distance varies as the square of the speed, we can set up the following equation based on the given information:
(d2 / d1) = (v2^2 / v1^2)
Substituting the given values:
(d2 / 79) = (70^2 / 55^2)
To find d2, we can solve for it algebraically:
d2 = (79 * 70^2) / 55^2
By calculating this expression, we can find the stopping distance of the truck traveling at 70 mph.
5) To calculate the amount in the account after 5 years with different compounding frequencies, we can use the formula for compound interest:
A = P * (1 + r/n)^(nt)
Where:
A = the final amount in the account
P = the principal amount (initial investment)
r = the annual interest rate (as a decimal)
n = the number of times interest is compounded per year
t = the number of years
In this case, the principal amount (P) is $7000, the annual interest rate (r) is 5% (0.05 as a decimal), and the number of years (t) is 5.
We can calculate the amount in the account after 5 years with different compounding frequencies using the following formulas:
For semiannual compounding:
A_semiannual = P * (1 + r/2)^(2*t)
For quarterly compounding:
A_quarterly = P * (1 + r/4)^(4*t)
For monthly compounding:
A_monthly = P * (1 + r/12)^(12*t)
By substituting the given values into each formula and calculating the expressions, we can find the amount in the account after 5 years for each compounding frequency.