![]() Thus, the point we have found is a local minimum. The second derivative of this guy is strictly positive for positive s, implying the function is concave up for positive s. To do so you must take the second derivative. We'll end up with h = 2 * 5 2/3 *7 1/3 / sqrt(3).ĮDIT: It's a bit pedantic, but technically you have to make sure that it's a local minimum at the value of s that I've found. From there, we can easily find the height by substituting into our previous formula. Step 3: The volume of the given triangular prism base area × length 93 × 15 1353 cubic inches. Step 2: The length of the prism is 15 in. So its area is found using the formula, 3a 2 /4 3 (6) 2 /4 93 square inches. Solving for s, we get that s = (4*350) 1/3 = 2 * 5 2/3 * 7 1/3. Step 1: The base triangle is an equilateral triangle with its side as a 6. ![]() We want to find the minimum so we set SA' = 0. SA = 2(sqrt(3)/4)s 2 + 3sh (the first term is the 2 triangular parts and the second term is the three lateral, rectangular parts).Īs a function of s alone, we have SA = 2(sqrt(3)/4)s 2 + 4sqrt(3)350/s. This is equivalent to h = 4*350/(sqrt(3)s 2 ). Step 3: Find the area of the rectangular sides by multiplying the perimeter of a base triangle by the length of the prism: A ( b 1 + b 2 + b 3) l. V = (sqrt(3)/4)hs 2 = 350 cm 3 (I converted mL to cm 3 for ease). Then the area of the base is (sqrt(3)/4)s 2. Let s be the base of the triangle and h be the height. This is an ordinary optimization problem so it requires the use of basic calculus. 3) Use the formula: Perimeter of Base X Height. ∅ ∖ ∁ ↦ ↣ ∩ ∪ ⊆ ⊂ ⊄ ⊊ ⊇ ⊃ ⊅ ⊋ ⊖ ∈ ∉ ∋ ∌ ℕ ℤ ℚ ℝ ℂ ℵ ℶ ℷ ℸ □ To find the surface area of a prism, follow the 5 steps: 1) Determine the height of the prism. Re-read your post before hitting submit, does it still make sense.Show your work! Detail what you have tried and what isn't working.Use proper spelling, grammar and punctuation.Give context and details to your question, not just the equation.Help others, help you! How to ask a good question Asking for solutions without any effort on your part, is not okay. Beginner questions and asking for help with homework is okay. Post your question and outline the steps you've taken to solve the problem on your own. Do not use ChatGPT in a question or an answerĭon't just post a question and say "HELP".Do not solicit or offer payments to complete your assignments or tests.No cheating - do not post questions from exams, tests, midterms, etc.No post flooding - Limit your posts to 2 or 3 questions a day.Don't be a jerk - don't be obnoxious or rude.Homework policy - asking for help is okay, asking to be given the solution is not.Make your question clear and concise - include steps you have tried.Stay on topic - this subreddit is for math questions no how-to guides, or non math related questions.Explain your post - show your efforts and explain what you are specifically confused with. ![]() ![]() Today, the surface area of a triangular prism remains a fundamental principle in geometry and continues to serve as a crucial element in a multitude of practical applications. Their studies on triangles, parallelograms, and three-dimensional shapes have greatly influenced contemporary understanding of geometry and the surface area of various shapes, including triangular prisms. While there is no definitive historical account of the origin of the triangular prism or its surface area concept, it can be traced back to ancient Greece, where mathematicians like Euclid and Pythagoras laid the groundwork for modern geometry. Moreover, artists and designers frequently employ triangular prisms in their creations, making the knowledge of surface area invaluable for conceptualizing and executing their work. In packaging design, calculating the surface area of a triangular prism helps optimize material usage, reduce waste, and minimize costs. For instance, in construction and architecture, the surface area plays a role in determining the stability and strength of structures, as well as insulation and energy efficiency. Triangular prisms, like other three-dimensional shapes, have numerous real-life applications that make understanding their surface area essential. The concept of surface area has broad applications in various fields, including engineering, architecture, and design, where it is crucial to estimate material requirements, costs, and structural integrity. ![]() A triangular prism consists of two congruent triangles at the ends, known as bases, connected by three parallelogram-shaped lateral faces. The surface area of a triangular prism is a key concept in geometry that pertains to the total area covering the external faces of the three-dimensional shape. ![]()
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