Road tunnel This activity is about using a gaphical o algebaic method to solve poblems in eal contets that can be modelled using quadatic epessions. The fist poblem is about a oad tunnel. The infomation sheet shows how you could solve the poblem. CC Geald England Road tunnel infomation sheet A oad tunnel is designed to have a coss-section consisting of a ectangle of height metes below a semi-cicle as shown. The aea of the coss-section must be at least 16 m to allow adequate ventilation. The stength of the mateials used to suppot the tunnel suggests that the aea must be no moe than 3 m because of the dange of collapse. Think about Fom the woding of the infomation, do you think that aea values of eactly 16 m and eactly 3 m ae acceptable values? Befoe you ead any futhe, see if you can wok out how to solve the poblem on you own, and ty to do so. If you get stuck, ead on. Using the infomation You will need to use the values given fo the aea A in ode to find the adius. Fist you need a fomula fo the aea in tems of. The aea of the semi-cicle is 0.5. The aea of the ectangle is = 4. So the fomula fo the aea of the tunnel entance is A = 0.5 + 4 The poblem can now be solved eithe by dawing a gaph o by using algeba. m Nuffield Fee-Standing Mathematics Activity Road tunnel Student sheets Copiable page 1 of 5
A Gaphical method This method uses a gaph to find the minimum and maimum values of the adius, metes, and then the ange of possible oad widths. The gaph can be dawn in Ecel o on gaph pape. Think about What speadsheet fomulae can you use to daw a gaph of A against fo values of up to 4 metes? Ty this 1 Put headings (m) and A (m ) at the top of columns A and B. Use 0 as the fist value of. Ente 0 in cell A. Wite a speadsheet fomula in cell B to calculate the aea. Wite a speadsheet fomula in cell A3 to calculate the net value of. See fomulae below. A B C A B C 1 (m) A(m ) 1 (m) A(m ) 0 0 0 =0.5*PI()*A^+4*A 3 0.5 3 =A+0.5 Numeical values Fomulae Use fill down to complete the table as fa as = 4. The esults ae shown below, with the column fomatted to show 1 decimal place and the A column to show decimal places: A B C A B C 1 (m) A(m ) 1 (m) A(m ) 0.0 0.00 0.0 =0.5*PI()*A^+4*A 3 0.5.39 3 =A+0.5 =0.5*PI()*A3^+4*A3 4 1.0 5.57 4 =A3+0.5 =0.5*PI()*A4^+4*A4 5 1.5 9.53 5 =A4+0.5 =0.5*PI()*A5^+4*A5 6.0 14.8 6 =A5+0.5 =0.5*PI()*A6^+4*A6 7.5 19.8 7 =A6+0.5 =0.5*PI()*A7^+4*A7 8 3.0 6.14 8 =A7+0.5 =0.5*PI()*A8^+4*A8 9 3.5 33.4 9 =A8+0.5 =0.5*PI()*A9^+4*A9 10 4.0 41.13 10 =A9+0.5 =0.5*PI()*A10^+4*A10 Numeical values Fomulae 3 Use the values in columns A and B to daw a gaph (select the cuve with points in the Scatte option). Add a title and labels. Nuffield Fee-Standing Mathematics Activity Road tunnel Student sheets Copiable page of 5
4 Include both majo and mino gidlines. Fomat them so that it is easy to ead off values of A and. Think about... how to use the gaph to find the maimum and minimum values of. 5a The gaph can be used to find the minimum value of, using the minimum value of 16 m fo the aea of the tunnel entance. The minimum value of is about.16 metes as shown below. Check this on you gaph. The oad width is. What is the minimum oad width? 5b The maimum aea is 3 m. Use the gaph to find the maimum value of, then the maimum oad width. Daw lines on you gaph to show the maimum and minimum values of. You can pint you gaph then do this by hand, o use the dawing tools in Ecel befoe pinting the gaph. Nuffield Fee-Standing Mathematics Activity Road tunnel Student sheets Copiable page 3 of 5
B Algebaic method The fomula fo the aea of the tunnel entance is A = 0.5 + 4 Fist the adius that will give an aea of 16 m can be found as shown below: Fo an aea of 16 m 0.5 + 4 = 16 This eaanges to 0.5 + 4 16 = 0 In the quadatic fomula: a = 0.5, b = 4, c = 16 Using the fomula: 4 4 4 4 0.5 16 0.5 116.53 = 4 10.795 Quadatic fomula Solutions of a + b + c = 0 ae b b 4ac a 4 10.795 6.795 The adius must be positive so. 163 This gives a minimum value fo the adius of.163 metes. The minimum tunnel width = =.163 = 4.33 The minimum width of the oad tunnel is 4.3 metes (to 1 decimal place). Ty this Use a simila method to find the value of that will give an aea of 3 m. Use this value of to find the maimum width of the tunnel. Ty these 1 The sketch shows the coss-section of a design fo a waste skip. a Show that the aea of the coss-section is given by the fomula A = + 0., whee is the length of the base and height in metes. b In ode that the skip should have the equied volume, the cosssectional aea must be.5 squae metes. Find the value of. 0. m 0. m Nuffield Fee-Standing Mathematics Activity Road tunnel Student sheets Copiable page 4 of 5
A containe is to be in the shape of a cylinde of height 1 cm. a Eplain why the total suface aea of the containe is given by: A = + 4 b The manufactue wants to limit the suface aea of the containe to 300 cm. Find the maimum adius. 1 cm 3 The fomula fo the volume of a bucket of height h is 1 V h R R whee R and ae the adii of the ends. 3 a A bucket is designed to be 4 cm high and to have a top with adius 13.5 cm. Show that fo this bucket: V 8 18.5 13.5 b The bucket is equied to have a volume of 10 lites, whee 1 lite = 1000 cm 3. Find the adius of the bottom of the bucket. 4 cm 13.5 cm 4 The sketch shows the coss-section of a wedge. Find the value of that would give a coss-sectional aea of: a 150 cm b 75 cm 4 cm 5 A quate cicle is to be emoved fom a ectangula metal plate to give the shape shown in the sketch. a Show that the emaining aea is A = 10 + 1 0.5 b It is equied that the aea should be 140 cm. Find the value of. cm 1 cm 10 cm At the end of the activity Descibe the steps in the gaphical method. Descibe the steps in the algebaic method. Which of the methods do you think is easie? Why? Was it the same fo evey question? If you found any pat of the questions difficult, what could you do to impove you mathematical skills? Nuffield Fee-Standing Mathematics Activity Road tunnel Student sheets Copiable page 5 of 5