{"id":4313,"date":"2024-08-05T08:40:00","date_gmt":"2024-08-05T12:40:00","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=4313"},"modified":"2024-10-25T09:06:00","modified_gmt":"2024-10-25T13:06:00","slug":"geometry-probability-2","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=4313","title":{"rendered":"Geometry Probability – 2"},"content":{"rendered":"\n

An iron rod of $1$ foot long is cut into three segments with random length. (1) What is the probability that the three segments form a triangle? (2) What is the probability that the three segments form an acute triangle? Click here<\/a> for the solution.<\/p>\n\n\n\n

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Solution<\/strong> (1): Let $x$, $y$, $z$ be the lengths of three segments. Therefore $$x+y+z=1 \\tag{1}$$ In order to form a triangle with these three segments, the following conditions must be met: $$x+y>z,y+z>x,z+x>y \\tag{2}$$<\/p>\n\n\n\n

Combining $(1)$ and $(2)$, we have $$x+y>\\dfrac{1}{2},x+y<1,x<\\dfrac{1}{2},y<\\dfrac{1}{2}\\tag{3}$$<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

In the above diagram, $\\triangle{ODE}$ represents all possible $x$ and $y$ values. And $\\triangle{ABC}$ represents all $x$ and $y$ values satisfying $(3)$. Therefore, the probability to the original question is $\\boxed{\\dfrac{1}{4}}$.<\/p>\n\n\n\n

Solution<\/strong> (2): Following Solution <\/strong>(1), the following additional conditions must be satisfied for the triangle to be acute: $$x^2+y^2>z^2, x^2+z^2>y^2,y^2+z^2>x^2\\tag{4}$$<\/p>\n\n\n\n

Replacing $z=1-(x+y)$ in $(4)$, we have: $$(1-x)(1-y)>\\dfrac{1}{2}, (x+y)(1-x)<\\dfrac{1}{2}, (x+y)(1-y)<\\dfrac{1}{2}\\tag{5}$$<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

In the above diagram, the area bounded by three hyperbola curves contains all $x$ and $y$ values that form an acute triangle.<\/p>\n\n\n\n

The area bounded by line $AB$ and hyperbola $AB$ is $$Area(AB)=\\int_{0}^{\\frac{1}{2}}\\left\\{\\left(1-\\dfrac{1}{2(1-x)} \\right)-\\left(\\dfrac{1}{2}-x\\right)\\right\\} \\,dx=\\int_{0}^{\\frac{1}{2}}\\left(\\frac{1}{2}+x-\\dfrac{1}{2(1-x)}\\right)\\,dx$$ $$=\\dfrac{1}{2}\\left(x+x^2+\\log{(1-x)}\\right)\\Big\\rvert_{0}^{\\frac{1}{2}}=\\dfrac{3}{8}-\\dfrac{1}{2}\\log{2}$$<\/p>\n\n\n\n

The area bounded by line $AC$ and hyperbola $AC$, which is same as the area bounded by line $BC$ and hyperbola $BC$, is $$Area(AC)=Area(BC)=\\int_{0}^{\\frac{1}{2}}\\left\\{\\dfrac{1}{2}-\\left(\\dfrac{1}{2(1-x)}-x\\right)\\right\\}\\,dx=$$ $$=\\int_{0}^{\\frac{1}{2}}\\left(\\frac{1}{2}+x-\\dfrac{1}{2(1-x)}\\right)\\,dx=\\dfrac{3}{8}-\\dfrac{1}{2}\\log{2}=Area(AB)$$<\/p>\n\n\n\n

The area bounded by three hyperbola curves is $$\\dfrac{1}{8}-3\\left(\\dfrac{3}{8}-\\dfrac{1}{2}\\log{2}\\right)=\\dfrac{3}{2}\\log{2}-1$$<\/p>\n\n\n\n

Therefore, the probability of three segments that form an acute triangle is $$2\\cdot\\left(\\dfrac{3}{2}\\log{2}-1\\right)=\\boxed{3\\log{2}-2\\approx 0.08}$$<\/p>\n\n\n\n<\/div>\n","protected":false},"excerpt":{"rendered":"

An iron rod of $1$ foot long is cut into three segments with random length. (1) What is the probability that the three segments form a triangle? (2) What is the probability that the three segments form an acute triangle? … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false},"categories":[13,17,8],"tags":[],"_links":{"self":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/4313"}],"collection":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=4313"}],"version-history":[{"count":48,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/4313\/revisions"}],"predecessor-version":[{"id":4554,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/4313\/revisions\/4554"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4313"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4313"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4313"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}