A Resource for Free-standing Mathematics Units

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1 A Resouce fo Fee-stnding Mthemtics Units A od tunnel is designed to hve coss section tht consists of m ectngle sumounted by semi-cicle s shown in the sketch. The height of the side of the tunnel is to be metes. The e of the coss section must be t lest 16 m to llow dequte ventiltion. The stength of the mteils used to suppot the tunnel suggests tht the e must be no moe thn 3 m becuse of the dnge of collpse. The designe needs to know how these constints ffect the width of the tunnel. In this ctivity you will use gph to find the minimum nd mimum vlues of the dius, metes, nd then the nge of possible od widths. Fist we need fomul fo the e in tems of. The e of the semi-cicle is 0.5 The e of the ectngle is = 4 So the fomul fo the e of the tunnel entnce is A = A gph cn be dwn in Ecel to show the e of the tunnel entnce fo diffeent vlues of. Follow these instuctions to dw the gph of A ginst fo vlues of up to 4 metes. How to do it. 1 Use the fomul to set up spedsheet tble. Put hedings (m) nd A (m ) t the top of columns A nd B. Use 0 s the fist vlue of. Ente 0 in cell A. Wite spedsheet fomul in cell B to clculte the e. Wite spedsheet fomul in cell A3 to clculte the net vlue of. See fomule below. A B C A B C 1 (m) A(m ) 1 (m) A(m ) =0.5*PI()*A^+4*A =A+0.5 Numeicl Fomule Use fill down to complete the tble s f s = 4. The esults e shown below with the column fomtted to show 1 deciml plce nd the A column to show deciml plces: A B C A B C 1 (m) A(m ) 1 (m) A(m ) =0.5*PI()*A^+4*A =A+0.5 =0.5*PI()*A3^+4*A =A3+0.5 =0.5*PI()*A4^+4*A =A4+0.5 =0.5*PI()*A5^+4*A =A5+0.5 =0.5*PI()*A6^+4*A =A6+0.5 =0.5*PI()*A7^+4*A =A7+0.5 =0.5*PI()*A8^+4*A =A8+0.5 =0.5*PI()*A9^+4*A =A9+0.5 =0.5*PI()*A10^+4*A Numeicl Fomule The Nuffield Foundtion 1 1

2 A Resouce fo Fee-stnding Mthemtics Units 3 To dw the gph, highlight the vlues in columns A nd B nd use the Cht Wizd to dw sctte gph (choose the cuves with points). When setting up the gph, wite in title nd lbels, emove the legend nd choose to use both mjo nd mino gidlines. 4 Fomt the gidlines so tht it is esy to ed off vlues of A nd. 5 The gph cn be used to find the minimum nd mimum vlues of : The e of the tunnel entnce should be t lest 16 m to llow dequte ventiltion. The minimum vlue of is bout.16 metes s shown below. Check this on you gph. The od width is. Wht is the minimum od width? 45 Ae of Coss Section Ae (m ) minimum Rdius (metes) b The mimum e is 3 m. Use the gph to find the mimum vlue of. Wht is the mimum od width? Dw lines on you gph to show the mimum nd minimum vlues of. You cn pint you gph then do this by hnd, o use the dwing tools in Ecel befoe pinting the gph. The Nuffield Foundtion

3 A Resouce fo Fee-stnding Mthemtics Units A od tunnel is designed to hve coss section tht consists of ectngle sumounted by semi-cicle s shown in the sketch. m The height of the side of the tunnel is to be metes. The e of the coss section must be t lest 16 m to llow dequte ventiltion, but the stength of the mteils used to suppot the tunnel suggests tht the e must be no moe thn 3 m becuse of the dnge of collpse. The designe needs to know how these constints ffect the width of the tunnel. The poblem cn be solved using qudtic equtions. Fist the dius tht will give n e of 16 m cn be found s shown below: How to do it The e of the semi-cicle is 0.5 nd the e of the ectngle is = 4 so the fomul fo the e of the tunnel entnce is A = Fo n e of 16 m = 16 This enges to = 0 In the qudtic fomul: Using the fomul: = 0.5, b = 4, c = 16 4 ± = 4 ± = = ± Qudtic Fomul Solutions of + b + c = 0 b ± b 4c e: = The dius must be positive so = = =. 163 This gives minimum vlue fo the dius of.163 metes. The minimum tunnel width = =.163 = The minimum width of the od tunnel is 4.3 metes (to 1 deciml plce). Use simil method to find the vlue of tht will give n e of 3 m Use this vlue of to find the mimum width of the tunnel. The Nuffield Foundtion 3 3

4 A Resouce fo Fee-stnding Mthemtics Units Now solve these: 1 The sketch shows the coss section of design fo wste skip. Show tht the e of the coss section is given by the fomul: A = + 0. whee is the length of the bse nd height in metes. 0. m 0. m b In ode tht the skip should hve the equied volume, the coss sectionl e must be.5 sque metes. Find the vlue of. A contine is to be in the shpe of cylinde of height 1 cm. 1 cm b Eplin why the totl sufce e of the contine is given by: A = + 4 The mnufctues wnt to limit the sufce e of the contine to 300 cm. Find the mimum dius. 3 The fomul fo the volume of bucket of height h is 1 V = h( R + R + ) whee R nd e the dii of the ends. 3 4 cm 13.5 cm A bucket is designed to be 4 cm high nd to hve top with dius 13.5 cm. Show tht fo this bucket: V = ( ) b The bucket is equied to hve volume of 10 lites whee 1 lite = 1000 cm 3. Find the dius of the bottom of the bucket. 4 The sketch shows the coss section of wedge. 4 cm Find the vlue of tht would give coss-sectionl e of: 150 cm b 75 cm 5 A qute cicle is to be emoved fom ectngul metl plte to give the shpe shown in the sketch. Show tht the emining e is A = cm b It is equied tht the e should be 140 cm. Find the vlue of. cm 10 cm The Nuffield Foundtion 4 4

5 A Resouce fo Fee-stnding Mthemtics Units Teche Notes Unit Intemedite Level, Using lgeb, functions nd gphs Notes This esouce gives two methods fo solving poblem involving qudtic function. You could use both methods togethe, o split the mteils nd use the two methods t diffeent times in the couse. The fist method involves using Ecel to dw qudtic gph. Students will need to be fmili with the use of functions nd fill down in Ecel nd lso know how to fomt gphs. The second method involves the use of the qudtic fomul. This pt of the esouce could be used s follow-up to nothe Nuffield esouce clled The Qudtic Fomul. The sheet of mied poblems cn be solved using eithe method. Answes Rod Using the Gph: Minimum od width is 4.3 m (to 1 dp). Mimum vlue of 3.4 metes, giving mimum od width of 6.8 metes (to 1 dp). Using the Qudtic Fomul: Fo n e of 3 m = 3 which enges to = 0 In the qudtic fomul: = 0.5, b = 4, c = 3 Using the fomul: 4 ± = ± = = 4 ± The dius must be positive so = = = The tunnel width = = = The mimum width is 6.8 metes (to 1 deciml plce). Othe Poblems. 1 b 1.5 m (to 1 dp) b 3. cm (to 1 dp) 3 b 9.4 cm (to 1 dp) cm (to 1 dp) b 11.6 cm (to 1 dp) cm (to 1 dp) The Nuffield Foundtion 5 5