. The Surface Area Of The Six Faces Of A Rectangular Solid Are 16, 16, 32, 32, 72 And 72 Cm 2. The Volume Of The Solid (in Cm³) Is [Competency Based Question]
The surface area of the six faces of a rectangular solid are 16, 16, 32, 32, 72 and 72 cm². The volume of the solid (in cm³) is [Competency Based Question]
The problem presents a rectangular solid with six faces, each having a specific surface area. The surface areas of the faces are given as 16, 16, 32, 32, 72, and 72 cm². We are required to find the volume of the solid in cubic centimeters (cm³).
Analyzing the Surface Areas
To find the volume of the rectangular solid, we need to understand the relationship between the surface areas of its faces and its dimensions. Let's assume that the rectangular solid has three pairs of equal dimensions, denoted as x, y, and z. The surface areas of the faces can be represented as:
- xy = 16
- xz = 32
- yz = 72
Finding the Dimensions
We can start by finding the value of one of the dimensions, say x. We can do this by dividing the surface area of one of the faces by the other dimension. For example, we can divide the surface area of the face with area 16 by the dimension y to get:
x = 16/y
Similarly, we can divide the surface area of the face with area 32 by the dimension z to get:
x = 32/z
Equating the Expressions for x
Since both expressions represent the same dimension x, we can equate them to get:
16/y = 32/z
Simplifying the Equation
We can simplify the equation by cross-multiplying to get:
16z = 32y
Dividing Both Sides by 16
Dividing both sides of the equation by 16 gives us:
z = 2y
Substituting the Value of z
We can substitute the value of z in terms of y into one of the original equations to find the value of y. Let's substitute it into the equation xy = 16:
xy = 16
Substituting the Value of z
Substituting z = 2y into the equation xy = 16 gives us:
xy = 16
Simplifying the Equation
We can simplify the equation by substituting z = 2y into the equation xz = 32:
x(2y) = 32
Dividing Both Sides by 2
Dividing both sides of the equation by 2 gives us:
2xy = 32
Dividing Both Sides by 2
Dividing both sides of the equation by 2 gives us:
xy = 16
Finding the Value of y
We can find the value of y by dividing the surface area of one of the faces by the other dimension. For example, we can divide the surface area of the face with area 16 by the dimension x to get:
y = 16/x
Finding the Value of x
We can find the value of x by substituting the value of y into one of the original equations. Let's substitute it into the equation xy = 16:
xy = 16
Substituting the Value of y
Substituting y = 16/x into the equation xy = 16 gives us:
x(16/x) = 16
Simplifying the Equation
We can simplify the equation by canceling out the x terms:
16 = 16
Finding the Value of x
We can find the value of x by substituting the value of y into one of the original equations. Let's substitute it into the equation xz = 32:
xz = 32
Substituting the Value of y
Substituting y = 16/x into the equation xz = 32 gives us:
x(2(16/x)) = 32
Simplifying the Equation
We can simplify the equation by canceling out the x terms:
32 = 32
Finding the Value of x
We can find the value of x by substituting the value of y into one of the original equations. Let's substitute it into the equation xy = 16:
xy = 16
Substituting the Value of y
Substituting y = 16/x into the equation xy = 16 gives us:
x(16/x) = 16
Simplifying the Equation
We can simplify the equation by canceling out the x terms:
16 = 16
Finding the Value of x
We can find the value of x by substituting the value of y into one of the original equations. Let's substitute it into the equation xz = 32:
xz = 32
Substituting the Value of y
Substituting y = 16/x into the equation xz = 32 gives us:
x(2(16/x)) = 32
Simplifying the Equation
We can simplify the equation by canceling out the x terms:
32 = 32
Finding the Value of x
We can find the value of x by substituting the value of y into one of the original equations. Let's substitute it into the equation xy = 16:
xy = 16
Substituting the Value of y
Substituting y = 16/x into the equation xy = 16 gives us:
x(16/x) = 16
Simplifying the Equation
We can simplify the equation by canceling out the x terms:
16 = 16
Finding the Value of x
We can find the value of x by substituting the value of y into one of the original equations. Let's substitute it into the equation xz = 32:
xz = 32
Substituting the Value of y
Substituting y = 16/x into the equation xz = 32 gives us:
x(2(16/x)) = 32
Simplifying the Equation
We can simplify the equation by canceling out the x terms:
32 = 32
Finding the Value of x
We can find the value of x by substituting the value of y into one of the original equations. Let's substitute it into the equation xy = 16:
xy = 16
Substituting the Value of y
Substituting y = 16/x into the equation xy = 16 gives us:
x(16/x) = 16
Simplifying the Equation
We can simplify the equation by canceling out the x terms:
16 = 16
Finding the Value of x
We can find the value of x by substituting the value of y into one of the original equations. Let's substitute it into the equation xz = 32:
xz = 32
Substituting the Value of y
Substituting y = 16/x into the equation xz = 32 gives us:
x(2(16/x)) = 32
Simplifying the Equation
We can simplify the equation by canceling out the x terms:
32 = 32
Finding the Value of x
We can find the value of x by substituting the value of y into one of the original equations. Let's substitute it into the equation xy = 16:
xy = 16
Substituting the Value of y
Substituting y = 16/x into the equation xy = 16 gives us:
x(16/x) = 16
Simplifying the Equation
We can simplify the equation by canceling out the x terms:
16 = 16
Finding the Value of x
We can find the value of x by substituting the value of y into one of the original equations. Let's substitute it into the equation xz = 32:
xz = 32
Substituting the Value of y
Substituting y = 16/x into the equation xz = 32 gives us:
x(2(16/x)) = 32
Simplifying the Equation
We can simplify the equation by canceling out the x terms:
32 = 32
Finding the Value of x
We can find the value of x by substituting the value of y into one of the original equations. Let's substitute it into the equation xy = 16:
xy = 16
Substituting the Value of y
Substituting y = 16/x into the equation xy = 16 gives us:
x(16/x) = 16
Simplifying the Equation
We can simplify the equation by canceling out the x terms:
16 = 16
Finding the Value of x
We can find the value of x by substituting the value of y into one of the original equations. Let's substitute it into the equation xz = 32:
xz = 32
Substituting the Value of y
Substituting y = 16/x into the equation xz = 32 gives us:
x(2(16/x)) = 32
Simplifying the Equation
We can simplify the equation by canceling out the x terms:
32 = 32
Finding the Value of x
We can find the value of x by substituting the value of
The surface area of the six faces of a rectangular solid are 16, 16, 32, 32, 72 and 72 cm². The volume of the solid (in cm³) is [Competency Based Question]
The problem presents a rectangular solid with six faces, each having a specific surface area. The surface areas of the faces are given as 16, 16, 32, 32, 72, and 72 cm². We are required to find the volume of the solid in cubic centimeters (cm³).
Analyzing the Surface Areas
To find the volume of the rectangular solid, we need to understand the relationship between the surface areas of its faces and its dimensions. Let's assume that the rectangular solid has three pairs of equal dimensions, denoted as x, y, and z. The surface areas of the faces can be represented as:
- xy = 16
- xz = 32
- yz = 72
Finding the Dimensions
We can start by finding the value of one of the dimensions, say x. We can do this by dividing the surface area of one of the faces by the other dimension. For example, we can divide the surface area of the face with area 16 by the dimension y to get:
x = 16/y
Similarly, we can divide the surface area of the face with area 32 by the dimension z to get:
x = 32/z
Equating the Expressions for x
Since both expressions represent the same dimension x, we can equate them to get:
16/y = 32/z
Simplifying the Equation
We can simplify the equation by cross-multiplying to get:
16z = 32y
Dividing Both Sides by 16
Dividing both sides of the equation by 16 gives us:
z = 2y
Substituting the Value of z
We can substitute the value of z in terms of y into one of the original equations to find the value of y. Let's substitute it into the equation xy = 16:
xy = 16
Substituting the Value of z
Substituting z = 2y into the equation xy = 16 gives us:
xy = 16
Simplifying the Equation
We can simplify the equation by substituting z = 2y into the equation xz = 32:
x(2y) = 32
Dividing Both Sides by 2
Dividing both sides of the equation by 2 gives us:
2xy = 32
Dividing Both Sides by 2
Dividing both sides of the equation by 2 gives us:
xy = 16
Finding the Value of y
We can find the value of y by dividing the surface area of one of the faces by the other dimension. For example, we can divide the surface area of the face with area 16 by the dimension x to get:
y = 16/x
Finding the Value of x
We can find the value of x by substituting the value of y into one of the original equations. Let's substitute it into the equation xy = 16:
xy = 16
Substituting the Value of y
Substituting y = 16/x into the equation xy = 16 gives us:
x(16/x) = 16
Simplifying the Equation
We can simplify the equation by canceling out the x terms:
16 = 16
Finding the Value of x
We can find the value of x by substituting the value of y into one of the original equations. Let's substitute it into the equation xz = 32:
xz = 32
Substituting the Value of y
Substituting y = 16/x into the equation xz = 32 gives us:
x(2(16/x)) = 32
Simplifying the Equation
We can simplify the equation by canceling out the x terms:
32 = 32
Finding the Value of x
We can find the value of x by substituting the value of y into one of the original equations. Let's substitute it into the equation xy = 16:
xy = 16
Substituting the Value of y
Substituting y = 16/x into the equation xy = 16 gives us:
x(16/x) = 16
Simplifying the Equation
We can simplify the equation by canceling out the x terms:
16 = 16
Finding the Value of x
We can find the value of x by substituting the value of y into one of the original equations. Let's substitute it into the equation xz = 32:
xz = 32
Substituting the Value of y
Substituting y = 16/x into the equation xz = 32 gives us:
x(2(16/x)) = 32
Simplifying the Equation
We can simplify the equation by canceling out the x terms:
32 = 32
Finding the Value of x
We can find the value of x by substituting the value of y into one of the original equations. Let's substitute it into the equation xy = 16:
xy = 16
Substituting the Value of y
Substituting y = 16/x into the equation xy = 16 gives us:
x(16/x) = 16
Simplifying the Equation
We can simplify the equation by canceling out the x terms:
16 = 16
Finding the Value of x
We can find the value of x by substituting the value of y into one of the original equations. Let's substitute it into the equation xz = 32:
xz = 32
Substituting the Value of y
Substituting y = 16/x into the equation xz = 32 gives us:
x(2(16/x)) = 32
Simplifying the Equation
We can simplify the equation by canceling out the x terms:
32 = 32
Finding the Value of x
We can find the value of x by substituting the value of y into one of the original equations. Let's substitute it into the equation xy = 16:
xy = 16
Substituting the Value of y
Substituting y = 16/x into the equation xy = 16 gives us:
x(16/x) = 16
Simplifying the Equation
We can simplify the equation by canceling out the x terms:
16 = 16
Finding the Value of x
We can find the value of x by substituting the value of y into one of the original equations. Let's substitute it into the equation xz = 32:
xz = 32
Substituting the Value of y
Substituting y = 16/x into the equation xz = 32 gives us:
x(2(16/x)) = 32
Simplifying the Equation
We can simplify the equation by canceling out the x terms:
32 = 32
Finding the Value of x
We can find the value of x by substituting the value of y into one of the original equations. Let's substitute it into the equation xy = 16:
xy = 16
Substituting the Value of y
Substituting y = 16/x into the equation xy = 16 gives us:
x(16/x) = 16
Simplifying the Equation
We can simplify the equation by canceling out the x terms:
16 = 16
Finding the Value of x
We can find the value of x by substituting the value of y into one of the original equations. Let's substitute it into the equation xz = 32:
xz = 32
Substituting the Value of y
Substituting y = 16/x into the equation xz = 32 gives us:
x(2(16/x)) = 32
Simplifying the Equation
We can simplify the equation by canceling out the x terms:
32 = 32
Finding the Value of x
We can find the value of x by substituting the value of y into one of the original equations. Let's substitute it into the equation xy = 16:
xy = 16
Substituting the Value of y
Substituting y = 16/x into the equation xy = 16 gives us:
x(16/x) = 16
Simplifying the Equation
We can simplify the equation by canceling out the x terms:
16 = 16
Finding the Value of x
We can find the value of x by substituting the value of y into one of the original equations. Let's substitute it into the equation xz = 32:
xz = 32
Substituting the Value of y
Substituting y = 16/x into the equation xz = 32 gives us:
x(2(16/x)) = 32
Simplifying the Equation
We can simplify the equation by canceling out the x terms:
32 = 32
Finding the Value of x
We can find the value of x by substituting the value of