{"id":182305,"date":"2025-01-13T16:10:51","date_gmt":"2025-01-13T16:10:51","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=182305"},"modified":"2025-01-13T16:10:53","modified_gmt":"2025-01-13T16:10:53","slug":"joint-variation-and-combined-variation-joint-variation-is-just-like-direct-variation","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/01\/13\/joint-variation-and-combined-variation-joint-variation-is-just-like-direct-variation\/","title":{"rendered":"Joint Variation and Combined Variation Joint variation is just like direct variation"},"content":{"rendered":"\n

Joint Variation and Combined Variation Joint variation is just like direct variation, but involves more than one other variable. All the variables are directly proportional, taken one at a time. Let\u00c3\u00a2\u00e2\u201a\u00ac\u00e2\u201e\u00a2s set this up like we did with direct variation, find the k, and then solve for y; we need to use the Formula Method: Joint Variation Problem Formula Method Suppose x varies jointly with y and the square root of z. When x=\u00c3\u00a2\u00cb\u2020\u00e2\u20ac\u212218 and y=2, then z=9. Find y when x=10 and z=4. x\u00c3\u00a2\u00cb\u2020\u00e2\u20ac\u212218\u00c3\u00a2\u00cb\u2020\u00e2\u20ac\u212218k=kyz\u00c3\u00a2\u00cb\u2020\u00c5\u00a1=k(2)9\u00c3\u00a2\u00e2\u201a\u00ac\u00e2\u20ac\u0153\u00c3\u00a2\u00cb\u2020\u00c5\u00a1=6k=\u00c3\u00a2\u00cb\u2020\u00e2\u20ac\u21223 xx1010y=kyz\u00c3\u00a2\u00cb\u2020\u00c5\u00a1=\u00c3\u00a2\u00cb\u2020\u00e2\u20ac\u21223yz\u00c3\u00a2\u00cb\u2020\u00c5\u00a1=\u00c3\u00a2\u00cb\u2020\u00e2\u20ac\u21223y4\u00c3\u00a2\u00e2\u201a\u00ac\u00e2\u20ac\u0153\u00c3\u00a2\u00cb\u2020\u00c5\u00a1=\u00c3\u00a2\u00cb\u2020\u00e2\u20ac\u21223y(2)=10\u00c3\u00a2\u00cb\u2020\u00e2\u20ac\u21226=\u00c3\u00a2\u00cb\u2020\u00e2\u20ac\u212253 Again, we can set it up almost word for word from the word problem. For the words \u00c3\u00a2\u00e2\u201a\u00ac\u00c5\u201cvaries jointly\u00c3\u00a2\u00e2\u201a\u00ac\u00c2\ufffd, just basically use the \u00c3\u00a2\u00e2\u201a\u00ac\u00c5\u201c=\u00c3\u00a2\u00e2\u201a\u00ac\u00c2\ufffd sign, and everything else will fall in place. Solve for k first by plugging in variables we are given at first; we get k=\u00c3\u00a2\u00cb\u2020\u00e2\u20ac\u21223. Now we can plug in the new values of x and z to get the new y. We see that y=\u00c3\u00a2\u00cb\u2020\u00e2\u20ac\u212253. Really not that bad! Joint Variation Word Problem: We know the equation for the area of a triangle is A=12bh (b= base and h= height), so we can think of the area having a joint variation with b and h, with k=12. Let\u00c3\u00a2\u00e2\u201a\u00ac\u00e2\u201e\u00a2s do an area problem, where we wouldn\u00c3\u00a2\u00e2\u201a\u00ac\u00e2\u201e\u00a2t even have to know the value for k: Joint Variation Problem Math and Notes The area of a triangle is jointly related to the height and the base. If the base is increased by 40% and the height is decreased by 10%, what will be the percentage change of the area? A=kbh(original)A=k(1.4b)(.9h)(new)A=k(1.4)(.9)bhA=k(1.26)bh Remember that when we increase a number by 40%, we are actually multiplying it by 1.4, since we have to add 40% to the original amount. Similarly, when we decrease a number by 10%, we are multiplying it by .9, since we are decreasing the original amount by 10%. Reduce the original values by the new values, and find the new \u00c3\u00a2\u00e2\u201a\u00ac\u00c5\u201cmultiplier\u00c3\u00a2\u00e2\u201a\u00ac\u00c2\ufffd; we see that there will be a 26% increase in the area (A would be multiplied by 1.26, or be 26% greater.) You can put real numbers to verify this, using the formula A=12bh. Joint Variation Word Problem: Here\u00c3\u00a2\u00e2\u201a\u00ac\u00e2\u201e\u00a2s another: Joint Variation Problem Math and Notes The volume of wood in a tree (V) varies directly as the height (h) and the square of the girth (g). If the volume of a tree is 144 cubic meters (m3) when the height is 20 meters and the girth is 1.5 meters, what is the height of a tree with a volume of 1000 and girth of 2 meters? V=k(height)(girth)2V=khg2144=k(20)(1.5)2=45k144=45k;k=3.2V=khg2;1000=3.2h\u00c3\u00a2\u00e2\u20ac\u00b9\u00e2\u20ac\u00a622h=78.125 We can set it up almost word for word from the word problem. For the words \u00c3\u00a2\u00e2\u201a\u00ac\u00c5\u201cvaries directly\u00c3\u00a2\u00e2\u201a\u00ac\u00c2\ufffd, just basically use the \u00c3\u00a2\u00e2\u201a\u00ac\u00c5\u201c=\u00c3\u00a2\u00e2\u201a\u00ac\u00c2\ufffd sign, and everything else will fall in place. Solve for k first; we get k=3.2. Now we can plug in the new values to get the new height. The new height is 78.125 meters. Question: 1. If quantity y is jointly related to quantity x and z, what happens to y as x and z increases? 2. If quantity y is jointly related to quantity x and z, what happens to y as x and z decreases? 3. What mathematical formulas model joint variation? 4. Complete the statement: If y=kxz then the quantity y varies _________jointly________________. K is called the __________constant________of variation. Joint\u00c3\u201a\u00c2  Variation and Combined Variation<\/p>\n\n\n\n

Joint variation<\/strong>\u00c3\u201a\u00c2 is just like direct variation, but involves more than one other variable. \u00c3\u201a\u00c2 All the variables are directly proportional, taken one at a time.<\/p>\n\n\n\n

Let\u00c3\u00a2\u00e2\u201a\u00ac\u00e2\u201e\u00a2s set this up like we did with direct variation, find the\u00c3\u201a\u00c2 k, and then solve for\u00c3\u201a\u00c2 y; we need to use the Formula Method:<\/p>\n\n\n\n

Joint Variation Problem<\/strong><\/td>Formula Method<\/strong><\/td><\/tr><\/thead>
Suppose\u00c3\u201a\u00c2 x\u00c3\u201a\u00c2 varies jointly with\u00c3\u201a\u00c2 y\u00c3\u201a\u00c2 and the square root of\u00c3\u201a\u00c2 z.\u00c3\u201a\u00c2 \u00c3\u201a\u00c2 When\u00c3\u201a\u00c2 x=\u00c3\u00a2\u00cb\u2020\u00e2\u20ac\u212218\u00c3\u201a\u00c2 and\u00c3\u201a\u00c2 y=2, then\u00c3\u201a\u00c2 z=9.\u00c3\u201a\u00c2 \u00c3\u201a\u00c2 Find\u00c3\u201a\u00c2 y\u00c3\u201a\u00c2 when\u00c3\u201a\u00c2 x=10\u00c3\u201a\u00c2 and\u00c3\u201a\u00c2 z=4.<\/td>x\u00c3\u00a2\u00cb\u2020\u00e2\u20ac\u212218\u00c3\u00a2\u00cb\u2020\u00e2\u20ac\u212218k=kyz\u00c3\u00a2\u00cb\u2020\u00c5\u00a1=k(2)9\u00c3\u00a2\u00e2\u201a\u00ac\u00e2\u20ac\u0153\u00c3\u00a2\u00cb\u2020\u00c5\u00a1=6k=\u00c3\u00a2\u00cb\u2020\u00e2\u20ac\u21223\u00c3\u201a\u00c2  \u00c3\u201a\u00c2  \u00c3\u201a\u00c2  \u00c3\u201a\u00c2  \u00c3\u201a\u00c2  \u00c3\u201a\u00c2  \u00c3\u201a\u00c2  \u00c3\u201a\u00c2  \u00c3\u201a\u00c2 \u00c3\u201a\u00c2 xx1010y=kyz\u00c3\u00a2\u00cb\u2020\u00c5\u00a1=\u00c3\u00a2\u00cb\u2020\u00e2\u20ac\u21223yz\u00c3\u00a2\u00cb\u2020\u00c5\u00a1=\u00c3\u00a2\u00cb\u2020\u00e2\u20ac\u21223y4\u00c3\u00a2\u00e2\u201a\u00ac\u00e2\u20ac\u0153\u00c3\u00a2\u00cb\u2020\u00c5\u00a1=\u00c3\u00a2\u00cb\u2020\u00e2\u20ac\u21223y(2)=10\u00c3\u00a2\u00cb\u2020\u00e2\u20ac\u21226=\u00c3\u00a2\u00cb\u2020\u00e2\u20ac\u212253\u00c3\u201a\u00c2 Again, we can set it up almost word for word from the word problem. For the words \u00c3\u00a2\u00e2\u201a\u00ac\u00c5\u201cvaries jointly\u00c3\u00a2\u00e2\u201a\u00ac\u00c2\ufffd, just basically use the \u00c3\u00a2\u00e2\u201a\u00ac\u00c5\u201c=\u00c3\u00a2\u00e2\u201a\u00ac\u00c2\ufffd sign, and everything else will fall in place.\u00c3\u201a\u00c2 Solve for\u00c3\u201a\u00c2 k\u00c3\u201a\u00c2 <\/strong>first by plugging in variables we are given at first; we get\u00c3\u201a\u00c2 k=\u00c3\u00a2\u00cb\u2020\u00e2\u20ac\u21223.\u00c3\u201a\u00c2 Now we can plug in the new values of\u00c3\u201a\u00c2 x\u00c3\u201a\u00c2 and\u00c3\u201a\u00c2 z\u00c3\u201a\u00c2 to get the new\u00c3\u201a\u00c2 y.\u00c3\u201a\u00c2 We see that\u00c3\u201a\u00c2 y=\u00c3\u00a2\u00cb\u2020\u00e2\u20ac\u212253. Really not that bad!<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

Joint Variation Word Problem:<\/strong><\/p>\n\n\n\n

We know the equation for the area of a triangle is\u00c3\u201a\u00c2 A=12bh\u00c3\u201a\u00c2 (b=\u00c3\u201a\u00c2 <\/em>base and\u00c3\u201a\u00c2 h=\u00c3\u201a\u00c2 height), so we can think of the area having a\u00c3\u201a\u00c2 joint variation<\/strong>\u00c3\u201a\u00c2 with\u00c3\u201a\u00c2 b\u00c3\u201a\u00c2 and\u00c3\u201a\u00c2 h, with\u00c3\u201a\u00c2 k=12. Let\u00c3\u00a2\u00e2\u201a\u00ac\u00e2\u201e\u00a2s do an area problem, where we wouldn\u00c3\u00a2\u00e2\u201a\u00ac\u00e2\u201e\u00a2t even have to know the value for\u00c3\u201a\u00c2 k:<\/p>\n\n\n\n

Joint Variation Problem<\/strong><\/td>Math and Notes<\/strong><\/td><\/tr><\/thead>
The area of a triangle is\u00c3\u201a\u00c2 jointly<\/strong>\u00c3\u201a\u00c2 related to the height and the base.\u00c3\u201a\u00c2 \u00c3\u201a\u00c2 If the base is increased by\u00c3\u201a\u00c2 40%<\/strong>\u00c3\u201a\u00c2 and the height is decreased by\u00c3\u201a\u00c2 10%<\/strong>, what will be the percentage change of the area?\u00c3\u201a\u00c2 <\/td>A=kbh(original)A=k(1.4b)(.9h)(new)A=k(1.4)(.9)bhA=k(1.26)bh\u00c3\u201a\u00c2 Remember that when we increase a number by\u00c3\u201a\u00c2 40%<\/strong>, we are actually multiplying it by\u00c3\u201a\u00c2 1.4<\/strong>, since we have to add\u00c3\u201a\u00c2 40%<\/strong>\u00c3\u201a\u00c2 to the original amount. Similarly, when we decrease a number by\u00c3\u201a\u00c2 10%<\/strong>, we are multiplying it by\u00c3\u201a\u00c2 .9<\/strong>, since we are decreasing the original amount by\u00c3\u201a\u00c2 10%<\/strong>.\u00c3\u201a\u00c2 Reduce the original values by the new values, and find the new \u00c3\u00a2\u00e2\u201a\u00ac\u00c5\u201cmultiplier\u00c3\u00a2\u00e2\u201a\u00ac\u00c2\ufffd; we see that there will be a\u00c3\u201a\u00c2 26%\u00c3\u201a\u00c2 increase<\/strong>\u00c3\u201a\u00c2 in the area (A\u00c3\u201a\u00c2 would be multiplied by\u00c3\u201a\u00c2 1.26<\/strong>, or be\u00c3\u201a\u00c2 26%<\/strong>\u00c3\u201a\u00c2 greater.)\u00c3\u201a\u00c2 You can put real numbers to verify this, using the formula\u00c3\u201a\u00c2 A=12bh.\u00c3\u201a\u00c2 <\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

Joint\u00c3\u201a\u00c2 Variation Word Problem:<\/strong><\/p>\n\n\n\n

Here\u00c3\u00a2\u00e2\u201a\u00ac\u00e2\u201e\u00a2s another:\u00c3\u201a\u00c2 <\/p>\n\n\n\n

Joint Variation Problem<\/strong><\/td>Math and Notes<\/strong><\/td><\/tr><\/thead>
The volume of wood in a tree (V) varies\u00c3\u201a\u00c2 directly<\/strong>\u00c3\u201a\u00c2 as the height (h) and the square of the girth (g).\u00c3\u201a\u00c2 \u00c3\u201a\u00c2 If the volume of a tree is\u00c3\u201a\u00c2 144<\/strong>\u00c3\u201a\u00c2 cubic meters (m3) when the height is\u00c3\u201a\u00c2 20<\/strong>\u00c3\u201a\u00c2 meters and the girth is\u00c3\u201a\u00c2 1.5<\/strong>\u00c3\u201a\u00c2 meters, what is the height of a tree with a volume of\u00c3\u201a\u00c2 1000<\/strong>\u00c3\u201a\u00c2 and girth of\u00c3\u201a\u00c2 2<\/strong>\u00c3\u201a\u00c2 meters?<\/td>\u00c3\u201a\u00c2 \u00c3\u201a\u00c2 V=k(height)(girth)2V=khg2144=k(20)(1.5)2=45k144=45k;k=3.2V=khg2;1000=3.2h\u00c3\u00a2\u00e2\u20ac\u00b9\u00e2\u20ac\u00a622h=78.125\u00c3\u201a\u00c2 We can set it up almost word for word from the word problem. For the words \u00c3\u00a2\u00e2\u201a\u00ac\u00c5\u201cvaries directly\u00c3\u00a2\u00e2\u201a\u00ac\u00c2\ufffd, just basically use the \u00c3\u00a2\u00e2\u201a\u00ac\u00c5\u201c=\u00c3\u00a2\u00e2\u201a\u00ac\u00c2\ufffd sign, and everything else will fall in place. Solve for\u00c3\u201a\u00c2 k\u00c3\u201a\u00c2 <\/strong>first; we get\u00c3\u201a\u00c2 k=3.2.\u00c3\u201a\u00c2 Now we can plug in the new values to get the new height.\u00c3\u201a\u00c2 The new height is\u00c3\u201a\u00c2 78.125<\/strong>\u00c3\u201a\u00c2 meters.<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

\u00c3\u201a\u00c2 <\/p>\n\n\n\n

Question:\u00c3\u201a\u00c2 <\/p>\n\n\n\n

1. If quantity y is jointly related to quantity x and z, what happens to y as x and z increases?<\/p>\n\n\n\n

2. If quantity y is jointly related to quantity x and z, what happens to y as x and z decreases?<\/p>\n\n\n\n

3. What mathematical formulas model joint variation?<\/p>\n\n\n\n

4. Complete the statement:<\/p>\n\n\n\n

If y=kxz then the quantity y varies _________jointly________________. K is called the __________constant________of variation.<\/p>\n\n\n\n

The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n

Joint variation occurs when a variable depends on the product or quotient of two or more other variables. In mathematical terms, if ( y ) varies jointly with ( x ) and ( z ), we express this relationship as:<\/p>\n\n\n\n

[ y = k \\cdot x \\cdot z ]<\/p>\n\n\n\n

where ( k ) is the constant of variation.<\/p>\n\n\n\n

1. Effect of Increasing ( x ) and ( z ) on ( y ):<\/strong><\/p>\n\n\n\n

If both ( x ) and ( z ) increase, ( y ) will also increase. This is because ( y ) is directly proportional to both ( x ) and ( z ). For example, if ( x ) and ( z ) each double, ( y ) will increase by a factor of four.<\/p>\n\n\n\n

2. Effect of Decreasing ( x ) and ( z ) on ( y ):<\/strong><\/p>\n\n\n\n

Conversely, if both ( x ) and ( z ) decrease, ( y ) will decrease. Since ( y ) is directly proportional to both variables, any reduction in ( x ) and ( z ) leads to a proportional decrease in ( y ).<\/p>\n\n\n\n

3. Mathematical Formulas Modeling Joint Variation:<\/strong><\/p>\n\n\n\n

The general formula for joint variation is:<\/p>\n\n\n\n

[ y = k \\cdot x \\cdot z ]<\/p>\n\n\n\n

This equation indicates that ( y ) varies jointly with ( x ) and ( z ). If ( y ) varies jointly with ( x ) and the square of ( z ), the formula becomes:<\/p>\n\n\n\n

[ y = k \\cdot x \\cdot z^2 ]<\/p>\n\n\n\n

These formulas can be adapted to include more variables or different powers, depending on the specific relationship.<\/p>\n\n\n\n

4. Completing the Statement:<\/strong><\/p>\n\n\n\n

If ( y = k \\cdot x \\cdot z ), then the quantity ( y ) varies jointly<\/strong>. ( k ) is called the constant<\/strong> of variation.<\/p>\n\n\n\n

Example Problem:<\/strong><\/p>\n\n\n\n

Suppose ( x ) varies jointly with ( y ) and the square root of ( z ). When ( x = -18 ), ( y = 2 ), and ( z = 9 ), find ( y ) when ( x = 10 ) and ( z = 4 ).<\/p>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

    \n
  1. Determine the constant ( k ):<\/strong> Using the initial conditions: [ x = k \\cdot y \\cdot \\sqrt{z} ] Substitute the known values: [ -18 = k \\cdot 2 \\cdot \\sqrt{9} ] [ -18 = k \\cdot 2 \\cdot 3 ] [ -18 = 6k ] [ k = -3 ]<\/li>\n\n\n\n
  2. Find ( y ) when ( x = 10 ) and ( z = 4 ):<\/strong> Use the formula with the new values: [ 10 = (-3) \\cdot y \\cdot \\sqrt{4} ] [ 10 = (-3) \\cdot y \\cdot 2 ] [ 10 = -6y ] [ y = -\\frac{10}{6} ] [ y = -\\frac{5}{3} ]<\/li>\n<\/ol>\n\n\n\n

    Therefore, when ( x = 10 ) and ( z = 4 ), ( y = -\\frac{5}{3} ).<\/p>\n\n\n\n

    For a visual explanation and additional examples on joint variation, you might find the following video helpful:<\/p>\n","protected":false},"excerpt":{"rendered":"

    Joint Variation and Combined Variation Joint variation is just like direct variation, but involves more than one other variable. All the variables are directly proportional, taken one at a time. Let\u00c3\u00a2\u00e2\u201a\u00ac\u00e2\u201e\u00a2s set this up like we did with direct variation, find the k, and then solve for y; we need to use the Formula Method: […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-182305","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/182305","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=182305"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/182305\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=182305"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=182305"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=182305"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}