Calculating Car Distance with Increasing Speed

This problem involves a car whose speed increases by a fixed amount each hour. We need to find out how many hours it will take for the car to cover a total distance of 435 km.

The car starts with a speed of 60 km/h. Each hour, the speed increases by 5 km/h. This means the distance covered in each successive hour forms an arithmetic progression.

• Initial speed (Speed in 1st hour) = 60 km/h
• Speed increase per hour = 5 km/h

Let's calculate the distance covered in each hour and the cumulative distance:

As shown in the table, the cumulative distance reaches exactly 435 km after 6 hours.

Alternatively, we can use the formula for the sum of an arithmetic series. The distance covered in the n-th hour is $D_n = 60 + (n-1)5$. The total distance after N hours is the sum $S_N = \sum_{n=1}^{N} D_n$.

This is an arithmetic series with first term $a_1 = 60$ and common difference $d = 5$. The sum of the first N terms is given by $S_N = \frac{N}{2}[2a_1 + (N-1)d]$.

We want to find N such that $S_N = 435$ km.

So, $435 = \frac{N}{2}[2(60) + (N-1)5]$

$435 = \frac{N}{2}[120 + 5N - 5]$

$435 = \frac{N}{2}[115 + 5N]$

Multiply both sides by 2:

$870 = N(115 + 5N)$

$870 = 115N + 5N^2$

Rearrange into a quadratic equation:

$5N^2 + 115N - 870 = 0$

Divide by 5:

$N^2 + 23N - 174 = 0$

Using the quadratic formula $N = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$:

$N = \frac{-23 \pm \sqrt{23^2 - 4(1)(-174)}}{2(1)}$

$N = \frac{-23 \pm \sqrt{529 + 696}}{2}$

$N = \frac{-23 \pm \sqrt{1225}}{2}$

$N = \frac{-23 \pm 35}{2}$

We get two possible solutions for N:

• $N_1 = \frac{-23 + 35}{2} = \frac{12}{2} = 6$
• $N_2 = \frac{-23 - 35}{2} = \frac{-58}{2} = -29$

Since the number of hours must be positive, we take $N = 6$.

Both methods show that the car will cover 435 km in 6 hours.