Julia Li Ladders This Investigation is to investigate why the width of alleyway doesn’t aﬀect the height ofintersection point of 2 ladders when the height that the ladders reach stays the same.The length of the alleyway is 250cm, ladder a reaches 280cm, ladder breaches 210cm. a bFirst I shall give a pre-exercise about the ladders.Finding the length of the ladders.The length of the ladders aﬀects the height that the ladder reaches.When the ladder leans on the side of the wall, they form a right-angledtriangle. Since the length of the ladder is the longest side of the triangleI shall use the Pythagoras Theorem formula to solve the length of theladder. (Pythagoras Theorem Formula)Ladder a: Ladder b: 375.37cm 325.58cmTherefore, I used the Pythagoras Theorem to find the length of the ladder.The length seems a bit inaccurate...so if imagining the x and y axes are shown on the diagram...thenext step shall be...Making equations for ladders.I used the linear equation (y=mx+c) to write the expressions of ladders.Ladder a: Ladder b: As you can see I multiplied every number by 100 at 4th step because, so I can convert the decimal points into whole numbers.
Julia LiDrawing the Line Graph.*Find the yellow line on Figure 1. It represents the line when the alleyway is 250cm wide.Intersection point: (142.8572, 120) 00The intersection point is 120cm away from the ground.Because our investigation is investigating how the width doesn’t aﬀect the height between theintersection point and ground. Therefore I’m going to find...Diﬀerent Widths.I’ll add another 3 widths, that will prove it. (300cm, 350cm, 400cm)If the width is 300cm...Ladder a: Ladder b: Key: Red Line Intersection Point: (171.4286, 120)If the width is 350cm...Ladder a: Ladder b: Key: Blue Line Intersection Point: (200, 120)If the width is 400cm...Ladder a: Ladder b: Key: Green Line Intersection Point: (228.5714, 120)Now I’m going to draw it on the graph, so I can see the pattern.
Julia Li (Figure 1) Key: 250cm width alleyway is the Yellow LineEquationsSince I got 4 examples, I will give one linear equation for each ladder that qualifies for all lines. Bysubstituting w as the width.Ladder a:Original Equation:Since 250 is the width, I simply substituted w to 250.Therefore:Ladder b:Original Equation:Since 250 is the width, I simply substituted w to 250 again.Therefore:But why is the height always the same?Explanation of Height of IntersectionBy logic, if the length of the ladder is able to be changed, the width doesn’t change the heightbetween intersection point and ground. Since the height that the ladder reaches is the same, andboth of them stays the same, the width doesn’t aﬀect the height between intersection point andground.But to give a better explanation, I will show you how the weight by extrapolating the formulas.
Julia LiThe intersection point of the two lines has to conform both of the formulas.If the intersection point =I can find .To find the , simply use both of the formulas. Because the intersection point has to conform bothof the formulas.Since I found the value of . Then I can find the value of .To find the value of , I substitute into one another formula again. I chose to do the one forLadder b.When I extrapolate the equation, you can see that the answer is 120 and the w is crossed out.Which proves us that the width doesn’t matter with the height of intersection.Now lets make a more universal equation, an equation that can be used for any height of ladders.Final Equation.Ladder a: the one that reaches 280cmLadder b: the one that reaches 210cmTherefore a=280, b=210 for substitutionThen I shall substitute it from the equation I made previously.
Julia LiTherefore the formula is:This formula can be used for any height that the ladder reaches.a & b are the height that the two ladders reaches. The formula is to find the height of theintersection point.