Question

A palm tree casts a shadow that is 7 m shorter than its height. The distance from the top of the tree to the tip of the shadow is 1 m longer than the tree itself. How tall is the palm tree?

Answer 1

Answer:

The palm tree is 12 m tall.

Step-by-step explanation:

1. The first to see is to consider this as triangle formed by the palm tree, the shadow, and the line from the top of the tree to the tip of the shadow to the tip the palm.

2. Assign variables to each distance in the triangle

Palm length= X

Shadow length= y

line from the top of the tree to the tip of the shadow to the tip the palm = Z

3. Describe the equation, reading the text:

(1) y=x-7

(2) z=x+1

(3) $z^{2} =x^{2} +y{2}$ (this last one is from pitagoras theorem)

4. We solve the system:

{y = x - 7, z = x + 1, z^2 = x^2 + y^2}

Solutions

x = 4, y = -3, z = 5

x = 12, y = 5, z = 13

Since the distance can't be negative. The answer is the palm tree is 12 meters tall.

https://www.wolframalpha.com/input/?i=y%3Dx-7%3B+z%3Dx%2B1%3Bz%5E2%3Dx%5E2%2By%5E2

Answer 2

Answer:

12 m

Step-by-step explanation:

Take a look at the picture below.

If we call x the height of the tree, then the shadow which is 7 m shorter than its height would be "x - 7" and the distance from the top of the tree to the tip of the shadow would be "x + 1" (it is 1 m longer than the tree)

• We can see that we have a triangle rectangle so we can use the Pythagorean theorem:

$(x+1)^{2} = x^{2} + (x-7)^{2} \\x^{2} +2x+1=x^{2} +x^{2} -14x+49\\x^{2} +2x+1=2x^{2} -14x+49\\0=x^{2} -16x+48\\0=(x-12)(x-4)$

• From this we get that x = 12 and x = 4
• However, if the height of the tree was 4, the shadow would be 4 - 7 = -3 which cannot be possible. Therefore, x = 4 is not a solution for this problem

Thus, the only solution we have is x = 12 and the height of the palm tree is 12 m.