QQuestionMathematics
QuestionMathematics
Find an expression for the volume of a triangular prism where the base of the prism is a right triangle with base 4x + 6 and height x + 5 and the height of the triangular prism is x + 5. Use the expression to find the volume when x = 3. Find an expression for the volume of a triangular prism.
Enter the correct expression in the box.
V(x)=
10 months agoReport content
Answer
Full Solution Locked
Sign in to view the complete step-by-step solution and unlock all study resources.
Step 1:: Recall the formula for the volume of a prism.
V(x) = \text{ (Area of the base right triangle)} imes \text{ (Height of the prism)}
A prism is a solid object with two identical faces, called bases, joined by a set of parallelograms. The volume V of a prism is given by the area of the base multiplied by the height of the prism. So, we have
Step 2:: Find the area of the right triangle.
A = \frac{1}{2} (4x + 6)(x + 5)
Here, the base of the triangle is given by 4x + 6 and the height is x + 5. Plug these values into the formula:
Step 3:: Expand the expression for the area.
A = \frac{1}{2} (4x^2 + 23x + 30)
Step 4:: Factor out the greatest common factor (GCF) from the expression.
A = \frac{1}{2} (x + 5)(4x + 6)
Step 5:: Substitute the area A into the volume formula.
V(x) = \frac{1}{2} (x + 5)(4x + 6)(x + 5)
Step 6:: Simplify the expression for the volume.
V(x) = \frac{1}{2} (x + 5)^2(4x + 6)
Final Answer
The volume of the triangular prism is given by the expression V(x) = \frac{1}{2} (x + 5)^2(4x + 6) To find the volume when x = 3, substitute x = 3 into the expression: V(3) = \frac{1}{2} (3 + 5)^2(4 \cdot 3 + 6) V(3) = \frac{1}{2} (8)^2(18) V(3) = 576 So, the volume of the triangular prism when x = 3 is 576 cubic units.
Need Help with Homework?
Stuck on a difficult problem? We've got you covered:
- Post your question or upload an image
- Get instant step-by-step solutions
- Learn from our AI and community of students