3.4 App. of quadratic functions

According to Ms. Goldfish’s research, the relationship between age([math]t[/math]) and the number of new events that can be remembered ([math]f(t)[/math]) can be represented as a quadratic function [math]f(t)=\displaystyle{\frac{-1}{13\cdot9}(t-25)^2+13}[/math].

(a) Mr. Worm is 38 years old. How many new events can he remember?(Integer)

(b) Base on the function, from how old will people remember nothing new?

Francis and Daniel went mountain climbing together. The figure shows the location of them. Let Francis’ location be [math](2,8)[/math] and Daniel’s location be [math](8,2)[/math]. Since Daniel is hungry, Francis threw a roast chicken leg to him, but the chicken leg was hijacked by an eagle at the highest point of the parabola. Assume the track function is a quadratic function and the height of the eagle is 10, find the quadratic function.

Happy Senior high school is holding a summer camp. Initially, 60 students were joining the camp. The fee was 5,000 Happy dollars for each student. To encourage more studnets to join the camp, the school modified the fee. For every student joining, the fee is reduced by 100 Happy dollars. Find the number of students joining the camp so that the revenue will be the highest.

Consider a parabolic tunnel with height 6.4 meters and width 8 meters. A double decker bus 2.5 meters wide and 4 meters heigh wants to pass the tunnel. Suppose the tunnel is two-way road. Can the bus pass the tunnel? Write “Yes” or “No”.

(Calculator) Mia wants to understand the IQ of her pet dog, Donut, because she found that it often does silly things. After conducting research, Mia uses a function [math]f(t)=-0.01(t-68)^2+62[/math] to gauge Donut’s IQ where [math]t[/math] represents Donut’s age in months.

(a) What is the IQ of Donut birth?

(b) In Donut’s life, how long, in months, does it take for its IQ to exceed 40? Round to integer.

Miria wants to host a solo dance performance. The price of one ticket is set at $20. To attract more audience, Miria decides that for every additional person beyond 50, the ticket price will decrease by $0.25.

(a) What is the audience number that maximizes the revenue?

(b) If Miria wants to earn at least $1000, what is the maximum number of audience members?

The shape of a tunnel is a two-way parabola. Its width is 10 meters and its height is 6 meters. In Happy City, the standard width of a truck is 2 meters. What is the height limit, in integer meters, of the tunnel for trucks?

(Calculator) On June 5, 1942, the famous Gustav Railway Gun officially participated in the Battle of Sevastopol, firing 800-millimeter caliber shells weighing 7 metric tons at the Soviet defensive positions. Let [math](0,0)[/math] be the coordinates of the railway gun’s base. Assuming the track function of the shell is [math]f(x)=-0.0003(x^2-25x)+0.0116[/math] (unit: km).

(a) Find the height, in meters, of the Gustav railway gun’s barrel (the initial position of the shell).

(b) Find the maximum height, in meters, reached by the shell during its trajectory. Round to 2 decimal places.

(c) If the shell misses any target , find the coordinates of its landing position. Round to 3 decimal places.

Ms. Cow sells pearl milk tea. She sells 50 cups of milk tea (40 NT each) every day. Since Ms. Cow is too tired recently, she wants to reduce the number of cups she sells. Therefore, she came up with a formula. Her formula is: For every cup reduced, the price is increased by 5 NT. How many cups should she sell to maximize her profit?

Hannah has a large piece of land, and she wants to build houses on them. By the rule of the market, if she builds one house, she will make 20 million; if she builds two houses, she will make 19 million from each house; if she builds three houses, she will make 18 million from each house, and so on. What is the maximum revenue, in billions, that she can earn?

(Calculator) There is a parabolic bridge connecting Happy city and Happy town. Suppose the distance between each pillar is the same. Find [math]x[/math] (meters).

Joe’s rectangular garden is 6 meters long and 4 meters wide. He wishes to double the area of his garden by increasing its length and width by the same amount. Find the number of meters by which each dimension must be increased.

A farmer wishes to construct a rectangular fenced area alongside a linear river of width [math]w[/math] and length [math]l[/math]. If ge needs no fence on the side with the river (i.e. only needs fence along three sides) and has 100ft of fence to work with and wants to enclose the largest area he can, what is the value of [math]w+l[/math]?

(Calculator) A bridge has a parabolic shape with span of 200m and a height of 50m. A bridge walk PQ is planned as a tourist attraction (35m above the road). PQ is supported by seven equally spaced supports.

Find the distance, in meters, between adjacent supports.

(Calculator) A cannonball is fired from a cliff that is 260 feet high with an initial speed of 128 feet per second. The height [math]s[/math] of the cannonball (in feet) as a function of time (in seconds) can be modeled by the function [math]s(t)=-16t^2+64t+260[/math].

(a) When will the height of the cannonball be 320 feet? If you have two or more answers, write one of them.

(b) When will the cannonball hit the ground?

(c) What is the highest point, in feet, of the cannonball?

(Calculator) The price for a new self-driving car made by company Alset can be modeled by the function [math]P(x)=-0.006x^2+50x[/math], where [math]P[/math] represents the price Alset can sell the car given [math]x[/math] number of buyers. If Alset hopes to sell the cat at $97,000, how many buyers should be? Round to the nearest integer. If you have two or moe answers, write the bigger one.

(Calculator) Andy is designing a rectangular dog pen for his dog. He plans to use 70 feet of fencing to enclose the area. The dog pen will be built along the wall of the house, so the side that borders the house does not need fencing. Additionally, he would divide the dog pen into three smaller rectangular areas with two fences that are parallel and congruent to the other two sides (see the figure). What is the maximum area that can be enclosed in this case? What are the dimensions of the greater rectangular area?

(a) What is the maximum enclosed area, in [math]\text{ft}^2[/math]?

(b) Find the length, in feet, of the greater rectangular area when the enclosed area is maximized.

To defend the Byzantine Empire, its king created a secret weapon called the Greek Fire. A defender spotted an enemy warship on the sea, so he shot an arrow with Greek Fire on it and destroyed the ship. Suppose the track function of the arrow is a quadratic function which has a vertex of (4,16) and the location of the defender is (0,12). Find the track function of the arrow.

Happy school is hosting an all-Upper-School Halloween Party at the Happy Amusement Park (HAP). Happy school hired a local tour company that charges 480 NTD per person. The service includes round trip bus rides between Happy school and the HAP and the cost of admissions to the park. Normally a ticket is 480 NTD per person, but for each person attending the party in excess of 60 people, the tour company will lower the ticket price for everyone by 1 NTD. The event will be canceled if there are less than 60 students who attend, and Happy school will only accept up to 250 students. Let [math]x[/math] be the number of students and [math]C(x)[/math] be the total cost that will be paid by Happy school budget.

(a)  What is the domain of [math]C(x)[/math]?

(b) If Happy school spends 70,000 NTD, what is the maximum number of students who can join the party?

Company Happy is designing a rectangular touch-screen TV and its diagonal should be 75 inches long. In addition, the length of the TV should be 6 inches longer than the width. Find the dimension of this TV and round to the nearest hundredth.