Algebra: chase rides his bike 5 miles to school in 25 minutes. the relationship between the distance he rides and the time it takes is a direct variation. a. write a direct variation equation between d, the distance chase rides, and the time it take, t. use k for the constant variation. b. use the information about chase's ride to school 5o find the value of k and it's units. c. if chase always rides at the same rate, how long will it take him to ride 8 miles?

(D) Distance traveled by Chase on bike = 5 miles

(T) Time taken by Chase to travel that distance = 25 minutes

D and T are directly related.

Part A (Equation)

Since, D and T are directly related,

⇒ D ∝ T (i.e. D is directly proportion to T)

⇒ D = k × T (using k as constant variation, as suggested in the question)

So the equation that represents the relation between T and D is D = k × T.

Part B (Value and units for k)

When we put the values of T and D in the equation D = k × T, we get:

5 = k × 25

Solving the above equation and moving 25 to the Left Hand Side (LHS) of the equation

⇒ 5 ÷ 25 = k

⇒ 0.2 = k OR

⇒ k = 0.2

Hence, value of k is 0.2. This is the rate at which Chase drove his bike to school.

Now, to determine the unit of k we will simply put the units of D and T in the equation.

⇒ 5 miles = k × 25 minutes

⇒ 0.2 miles/minute = k OR

⇒ k = 0.2 miles/minute

Hence, the unit for k is miles/minute or miles per minute.

Part C (Time taken to ride 8 miles)

Now, if Chase drives his bike at the same rate every day, i.e. at 0.2 miles/minute, then we need to determine how much time he takes to ride 8 miles. Putting the values of k and D in the equation:

⇒ 8 = 0.2 × T

⇒ T = 40 minutes

So, Chase would take 40 minutes to cover a distance of 8 miles.

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