Welcome to Westonci.ca, where you can find answers to all your questions from a community of experienced professionals. Connect with a community of experts ready to provide precise solutions to your questions quickly and accurately. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
Let's solve the given question by verifying each of the options against the known properties of the triangle and the law of cosines. The side lengths involved are [tex]\(a = 5\)[/tex], [tex]\(b = 7\)[/tex], and [tex]\(c = 10\)[/tex] with an angle of [tex]\(60^\circ\)[/tex] used for the calculations (since [tex]\(\cos(60^\circ) = 0.5\)[/tex]).
For Option A:
[tex]\[ b^2 = a^2 + c^2 - 2bc \cos(A) \][/tex]
Substituting the values we have:
[tex]\[ 7^2 = 5^2 + 10^2 - 2 \cdot 7 \cdot 10 \cdot \cos(60^\circ) \][/tex]
[tex]\[ 49 = 25 + 100 - 2 \cdot 7 \cdot 10 \cdot 0.5 \][/tex]
[tex]\[ 49 = 25 + 100 - 70 \][/tex]
[tex]\[ 49 = 125 - 70 \][/tex]
[tex]\[ 49 = 55 \][/tex]
Thus, the value obtained for this computation would be approximately [tex]\(55\)[/tex].
For Option B:
[tex]\[ b^2 = a^2 - c^2 - 2bc \cos(B) \][/tex]
Substituting the values:
[tex]\[ 7^2 = 5^2 - 10^2 - 2 \cdot 7 \cdot 10 \cdot \cos(60^\circ) \][/tex]
[tex]\[ 49 = 25 - 100 - 2 \cdot 7 \cdot 10 \cdot 0.5 \][/tex]
[tex]\[ 49 = 25 - 100 - 70 \][/tex]
[tex]\[ 49 = 25 - 170 \][/tex]
[tex]\[ 49 = -145 \][/tex]
The result for this computation is [tex]\(-145\)[/tex].
For Option C:
[tex]\[ b^2 = a^2 - c^2 - 2bc \cos(C) \][/tex]
Again, substituting the values:
[tex]\[ 7^2 = 5^2 - 10^2 - 2 \cdot 7 \cdot 10 \cdot \cos(60^\circ) \][/tex]
This computation will be exactly the same as the previous one (Option B), so:
[tex]\[ 49 = 25 - 100 - 70 \][/tex]
[tex]\[ 49 = -145 \][/tex]
So, the result for Option C is also [tex]\(-145\)[/tex].
For Option D:
[tex]\[ b^2 = a^2 + c^2 - 2ac \cos(B) \][/tex]
Substituting the values:
[tex]\[ 7^2 = 5^2 + 10^2 - 2 \cdot 5 \cdot 10 \cdot \cos(60^\circ) \][/tex]
[tex]\[ 49 = 25 + 100 - 2 \cdot 5 \cdot 10 \cdot 0.5 \][/tex]
[tex]\[ 49 = 25 + 100 - 50 \][/tex]
[tex]\[ 49 = 125 - 50 \][/tex]
[tex]\[ 49 = 75 \][/tex]
So, the value obtained here would be [tex]\(75\)[/tex].
Based on the results:
- Option A yields approximately [tex]\(55\)[/tex].
- Option B yields [tex]\(-145\)[/tex].
- Option C also yields [tex]\(-145\)[/tex].
- Option D yields [tex]\(75\)[/tex].
Comparing these with the numerical results:
- Option A: [tex]\(\approx 55\)[/tex]
- Option B: [tex]\(-145\)[/tex]
- Option C: [tex]\(-145\)[/tex]
- Option D: [tex]\(\approx 75\)[/tex]
We see that:
- Option A is correct: [tex]\(54.999999999999986 \approx 55\)[/tex]
- Option B is correct: [tex]\(-145.0\)[/tex]
- Option C is correct: [tex]\(-145.0\)[/tex]
- Option D is correct: [tex]\(74.99999999999999 \approx 75\)[/tex]
Thus, each option matches the results obtained, confirming the solutions are as listed from the given calculations.
For Option A:
[tex]\[ b^2 = a^2 + c^2 - 2bc \cos(A) \][/tex]
Substituting the values we have:
[tex]\[ 7^2 = 5^2 + 10^2 - 2 \cdot 7 \cdot 10 \cdot \cos(60^\circ) \][/tex]
[tex]\[ 49 = 25 + 100 - 2 \cdot 7 \cdot 10 \cdot 0.5 \][/tex]
[tex]\[ 49 = 25 + 100 - 70 \][/tex]
[tex]\[ 49 = 125 - 70 \][/tex]
[tex]\[ 49 = 55 \][/tex]
Thus, the value obtained for this computation would be approximately [tex]\(55\)[/tex].
For Option B:
[tex]\[ b^2 = a^2 - c^2 - 2bc \cos(B) \][/tex]
Substituting the values:
[tex]\[ 7^2 = 5^2 - 10^2 - 2 \cdot 7 \cdot 10 \cdot \cos(60^\circ) \][/tex]
[tex]\[ 49 = 25 - 100 - 2 \cdot 7 \cdot 10 \cdot 0.5 \][/tex]
[tex]\[ 49 = 25 - 100 - 70 \][/tex]
[tex]\[ 49 = 25 - 170 \][/tex]
[tex]\[ 49 = -145 \][/tex]
The result for this computation is [tex]\(-145\)[/tex].
For Option C:
[tex]\[ b^2 = a^2 - c^2 - 2bc \cos(C) \][/tex]
Again, substituting the values:
[tex]\[ 7^2 = 5^2 - 10^2 - 2 \cdot 7 \cdot 10 \cdot \cos(60^\circ) \][/tex]
This computation will be exactly the same as the previous one (Option B), so:
[tex]\[ 49 = 25 - 100 - 70 \][/tex]
[tex]\[ 49 = -145 \][/tex]
So, the result for Option C is also [tex]\(-145\)[/tex].
For Option D:
[tex]\[ b^2 = a^2 + c^2 - 2ac \cos(B) \][/tex]
Substituting the values:
[tex]\[ 7^2 = 5^2 + 10^2 - 2 \cdot 5 \cdot 10 \cdot \cos(60^\circ) \][/tex]
[tex]\[ 49 = 25 + 100 - 2 \cdot 5 \cdot 10 \cdot 0.5 \][/tex]
[tex]\[ 49 = 25 + 100 - 50 \][/tex]
[tex]\[ 49 = 125 - 50 \][/tex]
[tex]\[ 49 = 75 \][/tex]
So, the value obtained here would be [tex]\(75\)[/tex].
Based on the results:
- Option A yields approximately [tex]\(55\)[/tex].
- Option B yields [tex]\(-145\)[/tex].
- Option C also yields [tex]\(-145\)[/tex].
- Option D yields [tex]\(75\)[/tex].
Comparing these with the numerical results:
- Option A: [tex]\(\approx 55\)[/tex]
- Option B: [tex]\(-145\)[/tex]
- Option C: [tex]\(-145\)[/tex]
- Option D: [tex]\(\approx 75\)[/tex]
We see that:
- Option A is correct: [tex]\(54.999999999999986 \approx 55\)[/tex]
- Option B is correct: [tex]\(-145.0\)[/tex]
- Option C is correct: [tex]\(-145.0\)[/tex]
- Option D is correct: [tex]\(74.99999999999999 \approx 75\)[/tex]
Thus, each option matches the results obtained, confirming the solutions are as listed from the given calculations.
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.
