Answer:
The length of the shadow is increasing with the rate of 1.5 feet per sec
Explanation:
Let AB and CD represents the height of the lamppost and child respectively ( shown below )
Also, let E be a point represents the position of child.
In triangles ABE and CDE,
( right angles )
( common angles )
By AA similarity postulate,
![\triangle ABE\sim \triangle CDE](https://img.qammunity.org/2021/formulas/mathematics/high-school/idt3jh4fen0op73p2wfdn2cmvmfxvdeinv.png)
∵ Corresponding sides of similar triangles are in same proportion,
![\implies (AB)/(CD)=(BE)/(DE)](https://img.qammunity.org/2021/formulas/mathematics/high-school/lvthq8hrj8o2c3zycju7anlwvesd6clw71.png)
We have, AB = 12 ft, CD = 4 ft, BE = BD + DE = 6 + DE,
![\implies (12)/(4)=(6+DE)/(DE)](https://img.qammunity.org/2021/formulas/mathematics/high-school/q5pln2qbslmc5r40dw8mkibira8s31jbp4.png)
![12DE = 24 + 4DE](https://img.qammunity.org/2021/formulas/mathematics/high-school/mt6cpvrjswzw8vbb8lk8se23oy1f75whib.png)
![8DE = 24](https://img.qammunity.org/2021/formulas/mathematics/high-school/rr8urij20poal1izzvcc8kcrc25fhwgx2m.png)
![DE=3](https://img.qammunity.org/2021/formulas/mathematics/high-school/40c6avr81122riwo4o9nszaivqibzs2ea9.png)
Now, the speed of walking = 2 mph =
![(2* 5280)/(3600)\approx 2.933\text{ ft per sec}](https://img.qammunity.org/2021/formulas/mathematics/high-school/g2aai594dvr0zz6rj4c756i6v8rflsjyfn.png)
Note: 1 mile = 5280 ft, 1 hour = 3600 sec
Thus, the time taken by child to reach at E
![= \frac{\text{Walked distance}}{\text{Walking speed}}](https://img.qammunity.org/2021/formulas/mathematics/high-school/h1kxy89gmmlcm4vn5r02leditl3ospyhlo.png)
![=(6)/(2.933)](https://img.qammunity.org/2021/formulas/mathematics/high-school/7p7j48papgn0e211av7pqkegh7xt6y7ntd.png)
= 2.045 hours
Hence, the change rate in the length of shadow
![= \frac{\text{Length of shadow}}{\text{Time taken}}](https://img.qammunity.org/2021/formulas/mathematics/high-school/jweyocxcdupvnxw5uwj0zglmuqkhh7xe5p.png)
![=(3)/(2.045)](https://img.qammunity.org/2021/formulas/mathematics/high-school/d2e6i5zlpsegyxrg0tfeohgx1cja93jw88.png)
= 1.5 ft per sec.