Question

Chuck cycles along Skyline Drive.
He cycles 60 km at an average speed of x km/h.
He then cycles a further 45 km at an average speed of (r + 4)km/h.
His total joumey time is 6 hours.

(i) Write down an equation in rand show that it simplifies to 2r2-27r - 80 = 0.

(ii) Solve 2r2-27x - 80 = 0 to find the value of x.​

Answer 1

The equation that represents the situation is 2x²-27x - 80 = 0 and x value is 16

Given:

Chuck cycles along Skyline Drive.

He cycles 60 km at an average speed of x km/h.

He further cycles 45 km at an average speed (x+4)km/h  

His total journey time is 6 hours

To find:

(i) Write down an equation in x and show that it simplifies to 2x²-27x - 80 = 0.

(ii) Solve  2x²-27x - 80 = 0 for x

Solution:  

Complete Question:

Chuck cycles along Skyline Drive. He cycles 60 km at an average speed of x km/h. He then cycles a further 45 km at an average speed of (x + 4) km/h. His total journey time is 6 hours.  (i) Write down an equation in x and show that it simplifies to 2x²-27x - 80 = 0.

(ii) Solve  2x²-27x - 80 = 0 for x  

Formula used:

Time = Distance / Speed

Here given that

Chuck cycles 60 km at an average speed of x km/h  

=> Time taken to cycle 60 km = 60 km / x km/h = 60/x h    

He further cycles 45 km at an average speed (x+4)km/h  

=> Time taken to cycle 45 km = 45 km /(x+4)km/h = 45/(x+4) h

Given that the total time to journey = 6

=> $[\frac{60}{x} + \frac{45}{x+4} ] = 6$  

=> $\frac{60(x+4) + 45x}{x(x+4)} ] = 6$

=> $[\frac{60x+ 240 + 45x}{x^{2} +4x} ] = 6$          [ do cross multiplication ]

=> $60x+ 240 + 45x = 6(x^{2} +4x)$        

=> $60x+ 240 + 45x = 6x^{2} +24x$  

=> $6x^{2} +24x - 60x - 240 - 45x = 0$  [ add like terms ]

=> $6x^{2} - 81x -240 = 0$

=> $2x^{2} -27x - 80 = 0$     [ divided by 3 ]

Therefore,

The equation that represents the situation is 2x²-27x - 80 = 0

Now solve 2x²-27x - 80 = 0 as given below  

=> 2x² - 32x + 5x - 80 = 0    

=> 2x(x - 16) + 5(x - 16) = 0  

=>  (x - 16)(2x + 5) = 0  

=> x - 16 = 0                   2x = - 5

=> x = 16                           x = -5/2

Here we can't take speed with a negative sign

Therefore, x will be 16

Thus,

The equation that represents the situation is 2x²-27x - 80 = 0 and x value is 16

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Answer 2

Answer: Given equation is $2x^2-27x-80=0$ and value of x is 16.

Given :

1. Cover 60 km at an average speed of x km/h.
2. Cover 45 km at an average speed of (x + 4)km/h.
3. Total journey time is 6 hours.

To Find:

1. Write down an equation in terms of x.
2. Find the value of x.​

Step-by-step explanation:

For first part-

Step 1: As we know that total journey time is 6 hours. That means sum of time taken to cover both 60 km and 45 km is 6 hours.

Step 2: Time taken to complete a given distance = distance/speed

• Time taken to cover a distance of 60 km at an average speed of x km/h is $60/x$ hours.
• Time taken to cover a distance of 45 km at an average speed of (x+4) is $45/(x+4)$ hours.

Step 3: Total time taken = $\frac{60}{x}+\frac{45}{(x+4)}$

$6=\frac{60}{x}+\frac{45}{(x+4)}$

$6=\frac{60(x+4)+45x}{x(x+4)}\\6x(x+4)={60(x+4)+45x}}\\6x^2+24x=60x+240+45x\\6x^2-81x-240=0\\2x^2-27x-80=0$

For second part-

Step 1: Given equation is $2x^2-27x-80=0$

using the Quadratic Formula where

$a = 6, b = -81$ and $c = -240$

Now use Sridharacharya Formula to solve this quadratic equation.

$x = \frac{ -b \pm \sqrt{b^2 - 4ac}}{ 2a }\\\\x = \frac{ -(-81) \pm \sqrt{(-81)^2 - 4(6)(-240)}}{ 2(6) }\\\\x = \frac{ 81 \pm \sqrt{6561 - -5760}}{ 12 }\\\\x = \frac{ 81 \pm \sqrt{12321}}{ 12 }$

Step 2: The discriminant $b^2 - 4ac > 0$. So, there are two real roots.

$x = \frac{ 81 \pm 111\, }{ 12 }\\\\x = \frac{ 192 }{ 12 } \; \; \; x = -\frac{ 30 }{ 12 }\\\\x = 16 \; \; \; x = -\frac{ 5}{ 2 }$

The negative value of x is not possible. So the positive value of x which is 16 correct answer.

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