A lorry and a pick-up truck travel 180 km. The pick-up truck travels 20 km/h faster than the lorry. The…

Question

A lorry and a pick-up truck travel 180 km. The pick-up truck travels 20 km/h faster than the lorry. The pick-up truck arrives 45 minutes earlier than the lorry. Find the speeds of the lorry and pick-u

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Here's the solution to the problem:

Let the speed of the lorry be $x \text{ km/h}$ .
The speed of the pick-up is $20 \text{ km/h}$ faster than the lorry, so the speed of the pick-up is $(x+20) \text{ km/h}$ .
The distance covered by both vehicles is $180 \text{ km}$ .
The pick-up took $45 \text{ minutes}$ less than the lorry. Convert $45 \text{ minutes}$ to hours: $45 \text{ min} = \frac{45}{60} \text{ h} = \frac{3}{4} \text{ h}$ .

a) i) Write two expressions in terms of $x$ for the time taken by the lorry and the pick-up respectively to cover a distance of $180 \text{ km}$ .

Write the expression for the time taken by the lorry.
Time is calculated as $\frac{\text{Distance}}{\text{Speed}}$ .
$\text{Time taken by lorry} = \frac{180}{x} \text{ hours}$
Write the expression for the time taken by the pick-up.
$\text{Time taken by pick-up} = \frac{180}{x+20} \text{ hours}$
The expressions are:
Time for lorry: $\boxed{\frac{180}{x} \text{ hours}}$
Time for pick-up: $\boxed{\frac{180}{x+20} \text{ hours}}$

a) ii) Determine the speed of the lorry and that of the pick-up.
Set up the equation based on the time difference.
The pick-up took $\frac{3}{4}$ hours less than the lorry, so:
$\text{Time taken by lorry} - \text{Time taken by pick-up} = \frac{3}{4}$ $\frac{180}{x} - \frac{180}{x+20} = \frac{3}{4}$
Find a common denominator and combine the terms on the left side.
$\frac{180(x+20) - 180x}{x(x+20)} = \frac{3}{4}$ $\frac{180x + 3600 - 180x}{x^2 + 20x} = \frac{3}{4}$ $\frac{3600}{x^2 + 20x} = \frac{3}{4}$
Cross-multiply to eliminate the denominators.
$4 \times 3600 = 3(x^2 + 20x)$ $14400 = 3x^2 + 60x$
Rearrange the equation into a standard quadratic form $ax^2 + bx + c = 0$ .
$3x^2 + 60x - 14400 = 0$
Divide the entire equation by 3 to simplify.
$x^2 + 20x - 4800 = 0$
Solve the quadratic equation by factoring.
We need two numbers that multiply to $-4800$ and add to 20. These numbers are 80 and $-60$ .
$(x+80)(x-60) = 0$
Find the possible values for $x$ .
$x+80 = 0 \implies x = -80$ $x-60 = 0 \implies x = 60$
Since speed cannot be negative, we discard $x = -80$ .
Determine the speeds of the lorry and the pick-up.
Speed of the lorry ( $x$ ): $60 \text{ km/h}$
Speed of the pick-up ( $x+20$ ): $60 + 20 = 80 \text{ km/h}$

The speed of the lorry is $\boxed{60 \text{ km/h}}$ and the speed of the pick-up is $\boxed{80 \text{ km/h}}$ .

A lorry and a pick-up truck travel 180 km. The pick-up truck travels 20 km/h faster than the lorry. The pick-up truck arrives 45 minutes earlier than the lorry. Find the speeds of the lorry and pick-u

Step-by-step solution

Write the expression for the time taken by the lorry.

Write the expression for the time taken by the pick-up.

Set up the equation based on the time difference.

Find a common denominator and combine the terms on the left side.

Cross-multiply to eliminate the denominators.

Rearrange the equation into a standard quadratic form $ax^2 + bx + c = 0$ .

Divide the entire equation by 3 to simplify.

Solve the quadratic equation by factoring.

Find the possible values for $x$ .

Determine the speeds of the lorry and the pick-up.

Still stuck on this one?

More Mathematics Questions

A lorry and a pick-up truck travel 180 km. The pick-up truck travels 20 km/h faster than the lorry. The pick-up truck arrives 45 minutes earlier than the lorry. Find the speeds of the lorry and pick-u

Step-by-step solution

Write the expression for the time taken by the lorry.

Write the expression for the time taken by the pick-up.

Set up the equation based on the time difference.

Find a common denominator and combine the terms on the left side.

Cross-multiply to eliminate the denominators.

Rearrange the equation into a standard quadratic form ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0.

Divide the entire equation by 3 to simplify.

Solve the quadratic equation by factoring.

Find the possible values for xxx.

Determine the speeds of the lorry and the pick-up.

Still stuck on this one?

More Mathematics Questions

Rearrange the equation into a standard quadratic form $ax^2 + bx + c = 0$ .

Find the possible values for $x$ .