The Clown's Interceptor Pie

For the two pies to collied the x and y position have to be equal for the same time t .

First, find the time when the x positions are equal.

$100t = 500 - 500(t-2) \\ 100t = 500 - 500t + 1000 \\ 100t = 1500 - 500t \\ 600t = 1500 \\ t = {5 \over 2}$

The y positions have to be equal for this t value, so we can replace t and solve for K .

$80t - 16t^2 = K(t-2) - 16( t - 2)^2 \\ 80\cdot {5 \over 2} - 16 \cdot ({5 \over 2})^2 = K({5 \over 2} - 2) - 16({5 \over 2} - 2)^2 \\ 100 = {1 \over 2}K - 4 \\ 104 = {1 \over 2}K \\ K = \fbox{208}$

for can interceptor pie hit the pie thrown by Twinkle, need the two pies have same position in same time, so: $\left\{\begin{matrix} 100t = 500 - 500(t - 2) \\ 80t - 16t^2 = K(t-2) - 16(t-2)^2 \end{matrix}\right.$

$\left\{\begin{matrix} t = 5/2 \\ 80(5/2) - 16(5/2)^2 = K((5/2)-2) - 16((5/2)-2)^2 \end{matrix}\right.$

so $K = 208$

Let us take the two equations considering $x$ . We have $x=100t$ and $x=500-500(t-2)$ .

Substituting for x, we have $500-500(t-2)=100t$ . We can then solve for $t=\frac{5}{2}$ .

Now, we look at the two equations considering y. Plugging in $t=\frac{5}{2}$ , we have

$80(\frac{5}{2})-16(\frac{5}{2})^2=K(\frac{5}{2}-2)-16(\frac{5}{2}-2)^2$ . $200-100=K(\frac{1}{2})-16(\frac{1}{4})$ $100=\frac{K}{2}-4$

We can then simplify this down to $104=\frac{K}{2}$ , so we solve for $\boxed{K=208}$ .

For the two pies to meet at a point, the x & y co-ordinates must match after say t seconds. Hence, 100t = 500t - 500(t-2) giving t = 2.5 secs. Further, 80t - 16t^2 = K(t-2) -16(t-2)^2 where we plug in t = 2.5 to obtain K = 208

Setting the two expressions for the x values equal gives us t=2.5 Now we can plug that into the expression for y to get y= K (0.5) -16 (0.25) which also equals $80t - 16t^2$, and we can solve for k.

the first equation is supposed as "A" the second the third and the fourth equation as supposed as B, C, D. now simulatneously solving the A & C eqaution gives the value of t which is = 5/2. now simulataneously solving equation B & D gives the value of k when we put the value of t from the above equation in it as 208. which is the required answer

the Solution x = x 100 t = 500 - 500(t-2) 600 t = 1500 t=2,5 s

y=y 0,5k= 100+4 k=104/0,5 k=208