Answer:
Both ladder reaches 18.1 m up the building ⇒ 3rd answer
Step-by-step explanation:
* Lets study the information to solve the problem
- There are two ladders
- The lengths of them are 22 m and 20 m
- The bottom of the longer was 4 m farther than the bottom of the
shorter from the building
- Both of them reached the same distance up the building
* Lets solve the problem
- Let the distance between the bottom of the shorter ladder to the
building is x
∵ The bottom of the longer ladder is farther by 4
∴ The distance between the bottom of the longer ladder and the
building is x + 4
- Let the ladders reached the distance h up the building
* Now we have two right triangles
# Their hypotenuses are 22 and 20
# Their heights are h
# Their bases are x + 4 , x
- Lets find h in each triangle using the rule of Pythagoras
∵ (hypotenuse)² = (leg 1)² + (leg 2)²
# The longer ladder
∵ hypotenuse = 22
∵ leg 1 = x + 4
∵ leg 2 = h
∴ (22)² = (x + 4)² + h² ⇒ simplify
∴ 484 = (x + 4)² + h² ⇒ subtract (x + 4)² from both sides
∴ h² = 484 - (x + 4)² ⇒ (1)
# The shorter ladder
∵ hypotenuse = 20
∵ leg 1 = x
∵ leg 2 = h
∴ (20)² = (x )² + h² ⇒ simplify
∴ 400 = x² + h² ⇒ subtract x² from both sides
∴ h² = 400 - x² ⇒ (2)
- Equate (1) , (2) to find x
∴ 484 - (x + 4)² = 400 - x² ⇒ Add (x + 4)² and subtract 400 in both sides
∴ 84 = (x + 4)² - x² ⇒ open the bracket
∴ 84 = x² + 2(4)(x) + 4² - x² ⇒ simplify
∴ 84 = 8x + 14 ⇒ subtract 16 from both sides
∴ 68 = 8x ⇒ divide both sides by 8
∴ x = 8.5
- Substitute this value of x in (1) or (2) to find h
∵ h² = 400 - x²
∴ h² = 400 - (8.5)² = 327.75 ⇒ take √ for both sides
∴ h = 18.1
* Both ladder reaches 18.1 m up the building
![\bf \textit{Periodic/Cyclical Exponential Growth}\\\\ A=P(1 + r)^{\frac{t}{c}}\qquad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{initial amount}\to &650\\ r=rate\to 100\%\to \frac{100}{100}\to &1.00\\ t=\textit{elapsed time}\to &10\\ c=period\to &9 \end{cases} \\\\\\ A=650(1 + 1)^{\frac{10}{9}}\implies A=650(2)^{\frac{10}{9}}\implies A=650\sqrt[9]{2^{10}}](https://tex.z-dn.net/?f=%5Cbf%20%5Ctextit%7BPeriodic%2FCyclical%20Exponential%20Growth%7D%5C%5C%5C%5C%0AA%3DP%281%20%2B%20r%29%5E%7B%5Cfrac%7Bt%7D%7Bc%7D%7D%5Cqquad%20%0A%5Cbegin%7Bcases%7D%0AA%3D%5Ctextit%7Baccumulated%20amount%7D%5C%5C%0AP%3D%5Ctextit%7Binitial%20amount%7D%5Cto%20%26650%5C%5C%0Ar%3Drate%5Cto%20100%5C%25%5Cto%20%5Cfrac%7B100%7D%7B100%7D%5Cto%20%261.00%5C%5C%0At%3D%5Ctextit%7Belapsed%20time%7D%5Cto%20%2610%5C%5C%0Ac%3Dperiod%5Cto%20%269%0A%5Cend%7Bcases%7D%0A%5C%5C%5C%5C%5C%5C%0AA%3D650%281%20%2B%201%29%5E%7B%5Cfrac%7B10%7D%7B9%7D%7D%5Cimplies%20A%3D650%282%29%5E%7B%5Cfrac%7B10%7D%7B9%7D%7D%5Cimplies%20A%3D650%5Csqrt%5B9%5D%7B2%5E%7B10%7D%7D)