In an experiment, 26.5 g of metal was heated to 98.0°c and then quickly transferred to 150.0 g of water in a calorimeter. the initial temperature of the water was 20.0°c, and the final temperature after the addition of the metal was 32.5°c. assume the calorimeter behaves ideally and does not absorb or release heat.

This is a common laboratory experiment called calorimetry which determines the specific heat capacity of the sample metal. 

By the Conservation of Energy,

Energy of Metal = Energy of Water
mCmetalΔT = mCwaterΔT, wherein Cwater = 4.187 J/g·°C
(26.5 g)(Cmetal)(98 - 32.5°C) = (150 g)(4.187 J/g·°C)(32.5 - 20°C)
Solving for Cmetal,
Cmetal = 4.523 J/g·°C