If the McDonald's logo is constructed as $2$ parabolic curves of the form $y=-M{\left(x-6\right)}^2 \text{and }y=-M{\left(x+6\right)}^2$. If the McDonald's logo has a height to length ratio of $1.5$, then find the value of $M$.

Which identity is illustrated in the above image?

Akhil is an architect. He is into designing various villa projects. Akhil recently acquired $4$ acres of land to start a new villa project equipped with all necessary amenties. He decided to construct $10$ houses, each house occupying a land area of $25$ cents. He also designed small triangular-shaped parks in the plot for the kids to play. He also designed a beautiful garden in the middle of the plot.

When the construction of villas started, $20$ workers reached the plot. After the lunch, $288$ oranges are distributed among these $20$ men and women workers in such a way that each men get oranges which is same as the number of women and each women gets oranges which is same as double the number of men. If the number of women is maximum, find their numbers.

The main span of the Howrah bridge in Kolkata, West Bengal is about $194 \mathrm{m}$ long. The height of the towers is $26.2 \mathrm{m}$. The supporting cable for the main span of the bridge touches the road deck at the centre of the span. The parabola shown in the graph models the supporting cable of the main span of the Howrah Bridge and the green line segment represents the towers. Write down the values of $m, n, p$.

A tour bus in jaipur serves $400$ customers in a day. The charge is $₹50$ per person. The owner of bus service estimates that the company would lose $10$ passengers a day for each $₹5$ fare increase.

How much should the fare be in order to maximise the income for the company and find the maximum income for the company.

The diagram shows the graph of the function $y=A{x}^2+Bx+C$.

Complete the table below to show whether each expression is positive, negative or zero.

$\begin{array}{|llll|}\hline \text{ Expression }& \text{ positive }& \text{ negative }& \text{ zero }\\ A& & & \\ C& & & \\ B^2-4AC& & & \\ B& & & \\ \hline\end{array}$

Consider a person throwing a basketball $10 \text{m}$ above the ground. The basketball will reach its peak height and fall back to the ground. At time $t$, the height $h$ from the ground is given by $h=-2t^2+8t+10$. Calculate the time taken by the basketball to reach the ground.

When buying Embibe T-shirts, Raju determines that the total cost of buying $X$ T-shirts is given by the function,

$F\left(X\right)=0.04X^2-36X+9000$

If he wants to minimize the total cost of buying the T-shirts, how many shirts should he buy?