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QuestionMike initially stands 50 m50 \mathrm{~m} from a tree, covering it with his 7 cm7 \mathrm{~cm} finger. After moving back, his 6 cm6 \mathrm{~cm} finger covers the tree. How far did he move back?

Studdy Solution

STEP 1

Assumptions1. The length of Mike's middle finger is7 cm. The length of Mike's index finger is6 cm3. Initially, Mike is50 m away from the tree4. The tree's height is the same as the length of Mike's fingers when he holds them up to the tree5. The distance Mike moves backwards is what we need to find

STEP 2

We can use the concept of similar triangles to solve this problem. When Mike holds his finger up to the tree, the height of the tree and the length of his finger form two sides of a right triangle. The distance from Mike to the tree is the third side. Since the ratios of the sides of these triangles are the same, we can set up the following equationHeight of the treeInitial distance from the tree=Length of middle finger50\frac{Height\, of\, the\, tree}{Initial\, distance\, from\, the\, tree} = \frac{Length\, of\, middle\, finger}{50}

STEP 3

We can solve this equation for the height of the tree.
Height of the tree=Length of middle finger×501Height\, of\, the\, tree = \frac{Length\, of\, middle\, finger \times50}{1}

STEP 4

Plug in the value for the length of the middle finger to calculate the height of the tree.
Height of the tree=7×501Height\, of\, the\, tree = \frac{7 \times50}{1}

STEP 5

Calculate the height of the tree.
Height of the tree=350 cmHeight\, of\, the\, tree =350\, cm

STEP 6

Now, when Mike moves backwards, his index finger completely covers the tree. We can set up a similar equation as beforeHeight of the treeFinal distance from the tree=Length of index fingerFinal distance\frac{Height\, of\, the\, tree}{Final\, distance\, from\, the\, tree} = \frac{Length\, of\, index\, finger}{Final\, distance}

STEP 7

We can solve this equation for the final distance from the tree.
Final distance=Height of the tree×1Length of index fingerFinal\, distance = \frac{Height\, of\, the\, tree \times1}{Length\, of\, index\, finger}

STEP 8

Plug in the values for the height of the tree and the length of the index finger to calculate the final distance.
Final distance=3506Final\, distance = \frac{350}{6}

STEP 9

Calculate the final distance.
Final distance=58.33 mFinal\, distance =58.33\, m

STEP 10

Now that we have the final distance, we can find out how far Mike moved backwards by subtracting the initial distance from the final distance.
Distance moved backwards=Final distance−Initial distanceDistance\, moved\, backwards = Final\, distance - Initial\, distance

STEP 11

Plug in the values for the final distance and the initial distance to calculate the distance moved backwards.
Distance moved backwards=58.33−50Distance\, moved\, backwards =58.33 -50

STEP 12

Calculate the distance moved backwards.
Distance moved backwards=58.33−50=8.33 mDistance\, moved\, backwards =58.33 -50 =8.33\, mMike moved backwards by8.33 meters.

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