Question

nys common core mathematics curriculum lesson 6 problem set
name date

1. find the total volume of the figures, and record your solution strategy.

volume: volume:
solution strategy: solution strategy:
volume: volume:
solution strategy: solution strategy:
lesson 6:
find the total volume of solid figures composed of two
non - overlapping rectangular prisms.
date:
10/21/14

Explanation:

Step1: Calculate the volume of the first rectangular prism in part a

The formula for the volume of a rectangular prism is \(V = l\times w\times h\). For the lower part of figure a, \(l = 14\space cm\), \(w=3\space cm\), \(h = 5\space cm\). So \(V_1=14\times3\times5=210\space cm^3\).

Step2: Calculate the volume of the second rectangular prism in part a

For the upper part of figure a, \(l = 14\space cm\), \(w = 3\space cm\), \(h=6\space cm\) (since the total height is \(11\space cm\) and the lower height is \(5\space cm\)). So \(V_2=14\times3\times6 = 252\space cm^3\).

Step3: Sum the volumes for part a

The total volume \(V_a=V_1 + V_2=210+252 = 462\space cm^3\).

Step4: Calculate the volume of the first rectangular prism in part b

For the lower part of figure b, \(l = 15\space in\), \(w = 6\space in\), \(h = 4\space in\). So \(V_3=15\times6\times4=360\space in^3\).

Step5: Calculate the volume of the second rectangular prism in part b

For the upper part of figure b, \(l = 7\space in\), \(w = 6\space in\), \(h = 3\space in\). So \(V_4=7\times6\times3=126\space in^3\).

Step6: Sum the volumes for part b

The total volume \(V_b=V_3 + V_4=360+126=486\space in^3\).

Step7: Calculate the volume of the first rectangular prism in part c

For the lower part of figure c, \(l = 10\space cm\), \(w = 3\space cm\), \(h = 2\space cm\). So \(V_5=10\times3\times2 = 60\space cm^3\).

Step8: Calculate the volume of the second rectangular prism in part c

For the upper part of figure c, \(l=(10 - 6)=4\space cm\), \(w = 3\space cm\), \(h = 4\space cm\). So \(V_6=4\times3\times4=48\space cm^3\).

Step9: Sum the volumes for part c

The total volume \(V_c=V_5+V_6=60 + 48=108\space cm^3\).

Step10: Calculate the volume of the first rectangular prism in part d

For the lower part of figure d, \(l = 10\space m\), \(w = 3\space m\), \(h = 6\space m\). So \(V_7=10\times3\times6=180\space m^3\).

Step11: Calculate the volume of the second rectangular prism in part d

For the upper part of figure d, \(l = 8\space m\), \(w = 3\space m\), \(h=(12 - 6)=6\space m\). So \(V_8=8\times3\times6=144\space m^3\).

Step12: Sum the volumes for part d

The total volume \(V_d=V_7+V_8=180+144=324\space m^3\).

Answer:

a. Volume: \(462\space cm^3\), Solution Strategy: Split into two rectangular prisms and use \(V = l\times w\times h\) then sum.
b. Volume: \(486\space in^3\), Solution Strategy: Split into two rectangular prisms and use \(V = l\times w\times h\) then sum.
c. Volume: \(108\space cm^3\), Solution Strategy: Split into two rectangular prisms and use \(V = l\times w\times h\) then sum.
d. Volume: \(324\space m^3\), Solution Strategy: Split into two rectangular prisms and use \(V = l\times w\times h\) then sum.