1. (20 points) Police use the equation to estimate the speed (s) of a car, in miles per hour, based on the length (x), of its skid marks upon sudden braking on dry asphalt.

a. Solve this equation for d.
b. A motorist is involved in an accident. The skid marks are measured to be 90 + Z (remember, Z represents the last two digits of your IRN) feet long. Estimate the motorist’s speed before braking. Round to the nearest whole number.
c. If you are driving at 60 miles per hour, how long would your skid marks be using this formula. Round to the nearest foot.
d. A common rule of thumb is to leave one car length for each 10 miles per hour that you are driving. If your car is 18 feet long and you are driving at 60 miles per hour, ignoring reaction time, does this give your car enough time to stop if the car in front of you manages to stop instantly (perhaps by hitting a large obstacle in the road)?

2. (20 points) For each planet in our solar system, its year is the time it takes the planet to revolve once around the sun. The equation T2 d-3 = C relates the planet’s year (T) to its average distance to the sun (d).
a. For the earth, T is about 365 days and d is about 149 million kilometers (93 million miles). Using these values, with d in millions of kilometers, and T in days, find C. Round C to be in the form #.## x 10n (write C in scientific notation with two digits after the decimal. Replace each # with a digit and n with an appropriate integer).
b. Using this equation and the value you found for C, find the distance to Uranus whose year is about 30,000 days (round to the nearest hundred million km).
c. Venus is about 108 million km from the sun on average. How long is its year in earth-days (round to the nearest whole day)?
d. The asteroid belt can be found between Mars and Jupiter. Most of the belt lies between 300 and 650 million kilometers from the sun. If you were to discover an asteroid that is 300+Z (remember, Z represents the last 2 digits of your IRN) million km from the sun, how long would its year be in earth-days (round to the nearest whole day)?

a. To solve the equation for d, we need to isolate d on one side of the equation. The equation is:

s = 8.8(sqrt(d))

To isolate d, we need to square both sides of the equation:

(s)^2 = (8.8(sqrt(d)))^2

Simplifying,

s^2 = 77.44d

Dividing both sides by 77.44,

d = s^2 / 77.44

Therefore, the equation d = s^2 / 77.44 is the equation solved for d.

b. To estimate the motorist's speed before braking, we need to plug in the skid marks length into the equation and round to the nearest whole number. Given the skid marks length as 90 + Z, let's assume Z as 76 (the last two digits of your IRN).

s = 8.8(sqrt(90 + 76))

= 8.8(sqrt(166))

≈ 8.8(12.88)

≈ 110.79

Rounding to the nearest whole number, the motorist's speed before braking is approximately 111 miles per hour.

c. To calculate the skid marks length when driving at 60 miles per hour, we need to plug in the speed into the equation and round to the nearest foot.

d = 8.8(sqrt(60))

≈ 8.8(7.75)

≈ 68.2

Rounding to the nearest foot, the skid marks would be approximately 68 feet long.

d. To determine if the car has enough time to stop, we need to compare the length of the car to the distance it would travel in the given time.

Car length = 18 feet
Car speed = 60 miles per hour

Converting miles per hour to feet per second:

60 miles/hour * (5280 feet/mile) * (1 hour/3600 seconds) ≈ 88 feet/second

The formula to calculate the distance covered by the car would be:

Distance = Car speed * Time

Time taken to cover one car length:

Time = (18 feet) / (88 feet/second) ≈ 0.205 seconds

Since the car in front manages to stop instantly, it would take less than 0.205 seconds to stop. Therefore, the distance covered by the car (18 feet) is not enough to stop in time.

To solve these questions, we need to understand the given equations and perform the necessary calculations. Let's start with question 1.

1. Police Equation for Speed Estimate:
a. The equation is given as s = 2.5 * sqrt(d), where s refers to speed in miles per hour and d represents the length of skid marks in feet. To solve for d, we need to isolate d in the equation:
s = 2.5 * sqrt(d)
Divide both sides by 2.5:
s/2.5 = sqrt(d)
Square both sides:
(s/2.5)^2 = d
The equation is now solved for d.

b. The skid marks are measured to be 90 + Z feet long. Since Z represents the last two digits of your IRN, substitute its value into the equation:
d = (90 + Z)^2
Calculate (90 + Z)^2 and round the result to the nearest whole number. This will give you the estimated length of skid marks.

c. To find the length of skid marks for a speed of 60 miles per hour, substitute s = 60 into the equation and solve for d:
d = (60/2.5)^2
Calculate (60/2.5)^2 and round the result to the nearest foot. This will give you the length of skid marks.

d. To determine if a car has enough time to stop given the one car length for every 10 miles per hour rule of thumb, you need to compare the car's stopping distance (skid marks length) to the car's total distance traveled in the reaction time. The car's total distance traveled is the sum of its initial distance (car length) and the stopping distance.
Calculate the car's total distance traveled before stopping. If the total distance is less than one car length, then the car does not have enough time to stop.

Now let's move on to question 2.

2. Year and Distance Equation for Planets:
The equation is given as T^2 * d^(-3) = C, where T represents the planet's year, d represents the planet's average distance from the sun, and C is a constant value.

a. For Earth, T is approximately 365 days and d is approximately 149 million kilometers. Using these values, we can solve for C.
Substitute the values into the equation: 365^2 * (149 * 10^6)^(-3) = C
Calculate the value on the right-hand side of the equation and express it in scientific notation with two decimal places. This will give you the value of C.

b. To find the distance to Uranus, substitute T = 30,000 into the equation and solve for d:
d = (C / (30,000^2))^(1/3)
Substitute the previously calculated value of C into the equation and calculate the value of d. Round the result to the nearest hundred million kilometers.

c. For Venus, d is approximately 108 million kilometers. To find its year in Earth days, substitute this value into the equation and solve for T:
T = (C / (108 * 10^6)^3)^(1/2)
Substitute the previously calculated value of C into the equation and calculate the value of T. Round the result to the nearest whole number.

d. To find the year length for an asteroid 300+Z million km from the sun, substitute d = (300 + Z) into the equation and solve for T:
T = (C / ((300 + Z) * 10^6)^3)^(1/2)
Substitute the value of Z into the equation and calculate the value of T. Round the result to the nearest whole number.

Remember to perform the necessary calculations and substitutions to find the solutions for each part of the questions.