Chapter 11, Problem 38RP

a.

To determine

To calculate:Thearea ratio of region I and parallelogram ACDF .

Answer to Problem 38RP

Thearea ratio is $25:96$ .

Explanation of Solution

Given information:

AB : BC = $1:1$

FE : ED = $1:5$

Formula used:

Area of triangle: $A=\frac{1}{2}\ast b\ast h$

b = base of triangle

h = height of triangle Area of rectangle: $A=l\ast w$

l = length of rectangle

w= width of rectangle

Calculation:

  

Two triangles are similar by AA similarity rule.

  $\Delta BGC\sim \Delta DGE$

Therefore, the altitude is proportional to bases $\left(3:5\right)$

In region I, base $=5x$ and altitude $=5h$

In ACDF , base $=6x$ and altitude $=8h$

  $\begin{array}{l}\frac{A_{\mathrm{Ⅰ}}}{A_{▱}}=\frac{\left(\frac{1}{2}\ast b\ast h\right)}{\left(b\ast h\right)}\\ \frac{A_{\mathrm{Ⅰ}}}{A_{▱}}=\frac{\left(\frac{1}{2}\ast 5x\ast 5h\right)}{\left(6x\ast 8h\right)}\\ \frac{A_{\mathrm{Ⅰ}}}{A_{▱}}=\frac{\left(\frac{25xh}{2}\right)}{\left(48xh\right)}\\ \frac{A_{\mathrm{Ⅰ}}}{A_{▱}}=\frac{25}{96}\end{array}$

b.

To determine

To find: The probability that a landing in parallelogram ACDF would land in region II.

Answer to Problem 38RP

Theprobability is $\frac{47}{96}$ .

Explanation of Solution

Given information:

AB : BC = $1:1$

FE : ED = $1:5$

Formula used:

Area of triangle: $A=\frac{1}{2}\ast b\ast h$

b = base of triangle

h = height of triangle Area of rectangle: $A=l\ast w$

l = length of rectangle

w= width of rectangle

  $P\left(E\right)=\frac{Numberoffavourableoutcomes}{TotalNumberofoutcomes}$

Calculation:

  

Two triangles are similar by AA similarity rule.

  $\Delta BGC\sim \Delta DGE$

Therefore, the altitude is proportional to bases $\left(3:5\right)$

In region I, base $=5x$ and altitude $=5h$

In ACDF , base $=6x$ and altitude $=8h$

  $\begin{array}{l}{A}_{▱}=\left(b\ast h\right)\\ A_{▱}=\left(6x\ast 8h\right)\\ A_{▱}=48xh\end{array}$

  $\begin{array}{l}{A}_{\mathrm{Ⅱ}}=A_{▱}-A_{△ECD}-A_{△BGC}\\ A_{\mathrm{Ⅱ}}=\left(6x\ast 8h\right)-\frac{1}{2}(5x\ast 8h)-\frac{1}{2}(3x\ast 3h)\\ A_{\mathrm{Ⅱ}}=48xh-20xh-4.5xh\\ A_{\mathrm{Ⅱ}}=23.5xh\end{array}$

Probability means possibility.

The probability formula is defined as the possibility of an event to happen is equal to the ratio of the number of favorable outcomes and the total number of outcomes.

Probability of event to happen is

P(A landing in parallelogram ACDFwould land in region II) $=\frac{A_{\mathrm{Ⅱ}}}{A_{▱}}=\frac{23.5xh}{48xh}=\frac{47}{96}$