If `d=ut+(1)/(2)at^(2),u=20,a=10`, and d=50t, then find t.

Answer

To solve the equation \( d = ut + \frac{1}{2}at^2 \) with the given values \( u = 20 \), \( a = 10 \), and \( d = 50t \), we can follow these steps:

### Step 1: Substitute the values into the formula
We start with the formula:
\[ d = ut + \frac{1}{2}at^2 \]
Substituting the values of \( u \), \( a \), and \( d \):
\[ 50t = 20t + \frac{1}{2} \times 10 \times t^2 \]

### Step 2: Simplify the equation
Calculate \( \frac{1}{2} \times 10 \):
\[ \frac{1}{2} \times 10 = 5 \]
Now, substitute this back into the equation:
\[ 50t = 20t + 5t^2 \]

### Step 3: Rearrange the equation
To solve for \( t \), we need to rearrange the equation:
\[ 5t^2 + 20t - 50t = 0 \]
This simplifies to:
\[ 5t^2 - 30t = 0 \]

### Step 4: Factor the equation
We can factor out \( t \):
\[ t(5t - 30) = 0 \]

### Step 5: Solve for \( t \)
Setting each factor to zero gives us:
1. \( t = 0 \)
2. \( 5t - 30 = 0 \)

For the second equation:
\[ 5t = 30 \implies t = \frac{30}{5} = 6 \]

### Step 6: Determine the valid solution
Since time \( t \) cannot be zero in this context, we take the valid solution:
\[ t = 6 \]

Thus, the value of \( t \) is \( 6 \).

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