Question

Problem 1
Leni plans to use a ramp to make it easier to move a piano out of the back of his truck. The back of the truck is 33 inches tall and the ramp is 65 inches long. What is the horizontal distance from the end of the ramp to the back of the truck?

Problem 2
BongBong is admiring a statue in Seaside Park. The statue is 12 feet taller than he is, and BongBong is standing 16 feet away. How far is it from the top of the statue to BongBong's head

Problem 3
At ice-skating lessons, Manny attempts to do a complete revolution spin but only manages a quarter turn on her first attempt counterclockwise. What is the value of his first spin in degree angle?

Problem 4
A rectangular football field is 64 meters wide and 100 meters long. Ping runs from one corner of the field in a diagonal line to the opposite corner. How far did the player run?

Problem 5
The room thermostat was set to 22.034º C. Ronald wanted to set the temperature to 19.23º C. How many degrees’ cooler does he want it to be? Express your answer in minute-seconds.

Answer 1

Answer:

Let x be the purchased price.

Then the value after first year = x * 80/100 = 0.8x

The value after second year = 0.8x * 80/100 = 0 64x

Now we have, 0.64x = 315600

Hence purchased price = Rs. 493125

Now price after 3 years = 0 .64x * 80/100 * 80/100 * 80/100 = 0.64x * 0.512 = 0.32768x

Hence value of car, 3 years from now = 493125 * 0.32768 = Rs. 161587.20

Answer 2

Given:

Length of the back of truck = 33 inches

Length of ramp = 65 inches

Amount of length of statue taller than BongBong = 12 feet

Distance from which BongBong is standing away from the statue = 16 feet

Amount of turn Manny attempts to spin = $\frac{1}{4}$ turn counter-clockwise

Length of football field = 100m

Breadth of football field = 64m

Temperature set by thermostat = 22.034° C

Temperature that Ronald wants to set = 19.23° C

To find:

The horizontal distance from the end of the ramp to the back of the truck.

The distance from the top of the statue to BongBong's head.

The value of the first spin of Manny in degree angle.

Distance ran by the player.

Amount of cooling required for Ronald.

Solution:

1) The back of the truck is 33 inches long which represents a vertical line AB as shown in the figure and the ramp is a slanting line AC of length 65 inches.

We need to find the distance from the end of the ramp, i.e., point C to the back of the truck, i.e., point B. Let the back of the truck be ideally perpendicular to the ground, then ∠B = 90°. Hence, ΔABC is a right-angle triangle.

By Pythagoras theorem, the sum of squares of two sides of a right-angle triangle is equal to the square of the hypotenuse, i.e., in ΔABC,

$AC^{2}=AB^{2}+BC^{2}$

Here, AC is the hypotenuse and one of the sides AB is given. We need to find BC. Substituting the values in the above formula:

$65^{2} =33^{2}+BC^{2}$

$65^{2}-33^{2}=BC^{2}$

$4225-1089=BC^{2}$

$3136=BC^{2}$

$\sqrt{3136}=BC=56 inches$

Hence, the horizontal distance from the end of the ramp to the back of the truck is 56 inches.

2) The statue which BongBong admires in the park is 12 feet taller than him. Let this statue be ideally perpendicular to the ground. Let A be the point representing the top of the statue as shown in the figure. The horizontal distance between the statue and BongBong is given as 16 feet. Let BongBong be a small point B, compared to the huge size of the statue, as shown in the figure. The distance from the top of the statue to BongBong's head is given by AB. Since it is a right-triangle, by Pythagoras theorem,

$AB^{2}=BC^{2}+AC^{2}$

$AB^{2}=16^{2}+12^{2}$

$AB^{2}=256+144$

$AB^{2}=400$

$AB=20feet$

The distance from the top of the statue to BongBong's head is 20 feet.

3) One revolution means one full circle is completed. That means a spin of 360° is completed. But Manny manages a quarter turn in the counter-clockwise direction. On the $x-y$ plane, moving in the counter-clockwise direction means, moving in the positive $x-$direction. A quarter-turn of 360° means,

$\frac{1}{4}*360=90^{0}$

Thus, Manny has managed to spin 90° in his first attempt at ice-skating.

4) A rectangle is a two-dimensional shape having opposite sides equal and parallel. All of its angles are 90°. On drawing a diagonal from one corner to its opposite corner, we obtain two right triangles. Since the diagonal is the hypotenuse and the other two sides are given, by Pythagoras theorem,

$Diagonal^{2}=100^{2}+64^{2}$

$Diagonal^{2}=10000+4096$

$Diagonal^{2}=14096$

$Diagonal=118.72m$

In a rectangular football field of dimensions 100m×64m, Ping ran 118.72m in a diagonal manner.

5) Ronald wants the temperature to be lower than the set temperature of 22.034° C, which means he requires a much cooler temperature of 19.23°C. To determine how many degrees cooler he wanted, we simply subtract 19.23°C from 22.034° C.

Difference of temperature = $22.034-19.23=2.804^{0}C$

Thus, Ronald wanted to cool the room by 2.804° C.

The answers to the questions are:

1) The horizontal distance from the end of the ramp to the back of the truck is 56 inches.

2) The distance from the top of the statue to BongBong's head is 20 feet.

3) Manny has managed to spin 90° in his first attempt at ice-skating.

4) In a rectangular football field of dimensions 100m×64m, Ping ran 118.72m in a diagonal manner.

5) Ronald wanted to cool the room by 2.804° C.