This is a typical calorimetry question.

There are three heats involved:

#"heat lost by Cu + heat gained by water"color(white)(l) "+ heat gained by calorimeter" = 0#

#q_1 + q_2 + q_3 =0#

#m_1C_1ΔT_1 + m_2C_2ΔT_2 + C_3ΔT_3 = 0#

Let's calculate each heat separately.

#q_1 = m_1C_1ΔT_1 = 501 color(red)(cancel(color(black)("g"))) × "0.387 J"·color(red)(cancel(color(black)("g"^"-1")))"°C"^"-1" × (T_text(f) - "89.5 °C") = "193.9 J·°C"^"-1"(T_text(f) - "89.5 °C") = 193.9T_text(f) color(white)(l)"J·°C"^"-1" - "17 353 J"#

#q_2 = m_2C_2ΔT_2 = 159 color(red)(cancel(color(black)("g"))) × "4.184 J"·color(red)(cancel(color(black)("g"^"-1")))"°C"^"-1" × (T_text(f) - "22.8 °C") = "665.3 J·°C"^"-1"(T_text(f) - "22.8 °C") = 665.3T_text(f) color(white)(l)"J·°C"^"-1" - "15 168 J"#

#q_3 = C_3ΔT_3 = "10.0 J"·"°C"^"-1" × (T_text(f) - "22.8 °C") = 10.0T_text(f) color(white)(l)"J·°C"^"-1" - "228 J"#

Now, we add the three heats and combine like terms.

#q_1 + q_2 + q_3 = 193.9T_text(f) color(red)(cancel(color(black)("J")))·"°C"^"-1" - "17 353" color(red)(cancel(color(black)("J"))) + 665.3T_text(f) color(red)(cancel(color(black)("J")))·"°C"^"-1" - "15 168" color(red)(cancel(color(black)("J"))) +10.0T_text(f) color(red)(cancel(color(black)("J")))·"°C"^"-1" - 228 color(red)(cancel(color(black)("J"))) = 0#

#869.2 T_text(f)color(white)(l) "°C"^"-1" - "32 749" = 0#

#T_text(f) = "32 749"/("869.2 °C"^"-1") = "37.7 °C"#