Can someone help me understand why my text's solution for the value of a futures contract is different from how I would do it

I'm trying to solve this problem in the textbook, but the answer doesn't make sense to me given the formulas I learned. Here is the question itself:

A one-year long forward contract on a non-dividend-paying stock is entered into when the stock price is $40 and the risk-free rate of interest is 10% per annum with continuous compounding. (a) What are the forward price and the initial value of the forward contract? (b) Six months later, the price of the stock is $45 and the risk-free interest rate is still 10%. What are the forward price and the value of the forward contract?

The first part is simple. Futures price = spot price e^(rate time). This is 402.718^(.101) = 44.21. The initial value is 0.

The second part I don't understand. The formula I have for value of future price is:

Value of forward contract = (futures price - delivery price)e^(-rate * time)

However, the solution I've found says to do (45 - 44.21)e^(0.1*.5)

I dont understand this. Wouldn't 45 be the spot price, not the futures price? Didn't we already calculate the futures price as 44.21? So why are we using 45 as the futures price? And as for the formula itself, why would the futures price be different from the delivery price to begin with?

Thanks for any help