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3 years ago

6

How many times does the digit 7 appear among the terms of the sequence of consecutive integer numbers 7, 8, 9, ...., 777?

2 answers:

harina [27]3 years ago

7 0

Answer:

It appears 234 times

Step-by-step explanation:

For first 6 hundreds i.e, 17-107, 117-207, 217-307, 317-407, 417-507, 517-607, there are 20 sevens for each. Which gives a total of 20*6 = 120 sevens.

The number '7' along with this makes a total of 121 sevens.

From 617-707, there are 28 sevens. TOTAL = 121 + 28 = 149

From 708-716, there are 9 sevens.

Now, from 717-777, we have a total of 76 sevens

Adding these all makes total= 149+9+76 = 234

Dahasolnce [82]3 years ago

4 0

Answer:

234 times

Step-by-step explanation:

<u>Number of times the number 7 appears in a hundred</u>

7 as units digit (07-17-27 ..... 97): 10 times

7 as tens digit (70-71-72..... 79): 10 times

20 times the digit 7 appears in first one hundred (0-100)

Let's calculate how many times 7 would be as units or tens in 7 hundreds

20X7 = 140 times digit 7 appears until number 699

<u>Now, from 700 to 777</u>

7 as hundreds digit (700-701-702 .... 777): 78 times

7 as tens digit (770-771-772 .... 777): 8 times

7 as units digit (707-717-727....777): 8 times

78 + 8 + 8 = 94 times the digit 7 appears in the range 700 - 777. Plus 140 times

140 + 94 = 234 times

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2 years ago

The Colonel spots a campfire at a bearing N 59∘59∘ E from his current position. Sarge, who is positioned 242 feet due east of th

Rashid [163]

Answer:

i. Colonel is about 201 feet away from the fire.

ii. Sarge is about 125 feet away from the fire.

Step-by-step explanation:

Let the Colonel's location be represented by A, the Sarge's by B and that of campfire by C.

The total angle at the campfire from both the Colonel and Sarge = $59^{0}$ + $34^{0}$

= $93^{0}$

Thus,

<CAB = $90^{0}$ - $59^{0}$ = $31^{0}$

<CBA = $90^{0}$ - $34^{0}$ = $56^{0}$

Sine rule states;

$\frac{a}{Sin A}$ = $\frac{b}{Sin B}$ = $\frac{c}{Sin C}$

i. Colonel's distance from the campfire (b), can be determined by applying the sine rule;

$\frac{b}{Sin B}$ = $\frac{c}{Sin C}$

$\frac{b}{Sin 56^{0} }$ = $\frac{242}{Sin 93^{0} }$

$\frac{b}{0.8290}$ = $\frac{242}{0.9986}$

cross multiply,

b = $\frac{0.8290*242}{0.9986}$

= 200.8993

Colonel is about 201 feet away from the fire.

ii. Sarge's distance from the campfire (a), can be determined by applying the sine rule;

$\frac{a}{Sin A}$ = $\frac{c}{Sin C}$

$\frac{a}{Sin 31^{0} }$ = $\frac{242}{Sin 93^{0} }$

$\frac{a}{0.5150}$ = $\frac{242}{0.9986}$

cross multiply,

a = $\frac{0.5150*242}{0.9986}$

= 124.8073

Sarge is about 125 feet away from the fire.

8 0

3 years ago

Use the formula for the cosine of the difference of two angles to find the exact value of the following expression.

Zolol [24]

Answer:

Exact value of Cos(45° - 60°) is 0.96 using difference of two angles.

Step-by-step explanation:

Given:

Cos(45° - 60°)

We have to apply the formula of cosine for difference of the two angles.

Formula:

$cos(a-b)=cos(a)\ cos(b)+sin(a)\ sin(b)$

Plugging the values.

⇒ $cos(45-60)=cos(45)\ cos(60) + sin(45)\ sin(60)$

We know that the values :

$sin(45) =cos(45) = \frac{1}{\sqrt{2} }$

$sin(60)=\frac{\sqrt{3} }{2}$ and $cos(60)=\frac{1}{2}$

So,

⇒ $cos(45-60)=(\frac{1}{\sqrt{2} } \times \frac{1}{2} ) + (\frac{1}{\sqrt{2} } \times \frac{\sqrt{3} }{2})$

⇒ $cos(45-60)=(\frac{1}{2\sqrt{2} } + \frac{\sqrt{3} }{2\sqrt{2} })$

⇒ $cos(45-60)=(\frac{1+\sqrt{3} }{2\sqrt{2} } )$

⇒ $cos(45-60)=(\frac{1+\sqrt{3} }{2\sqrt{2} } )\times \frac{2\sqrt{2} }{2\sqrt{2} }$ ...<em>rationalizing </em>

⇒ $cos(45-60)=\frac{2\sqrt{2} +2\sqrt{6} }{ 8}$

⇒ $cos(45-60)=\frac{2(\sqrt{2}+\sqrt{6})}{8}$ ...<em>taking 2 as a common factor</em>

<em>⇒ </em> $cos(45-60)=\frac{(\sqrt{2}+\sqrt{6})}{4}$

To find the exact values we have to put the values of sq-rt .

As<em>, </em> $\sqrt{2}=1.41$ and $\sqrt{6} =2.44$

Then

<em>⇒ </em> $cos(45-60)=\frac{( 1.41+2.44)}{4}$ <em />

<em>⇒ </em> $cos(45-60)=\frac{( 3.85)}{4}$ <em />

⇒ $cos(45-60)=0.96$

So the exact value of Cos(45° - 60°) is 0.96 using difference of two angles.

3 0

3 years ago