k. (64 to the root of 3)^4
l. (3 to the root of 4)^-8
f. (5^2/5/^2)^-1/2
4. The radius of a sphere r is given in terms of its volume V by the formula: r=(0.75V/π)^1/3 By how many inches has the radius of a spherical balloon increased when tthe amount of air in the ballon is increased from 4.2ft^3 to 24.9ft^3?
2. The gas used by a car on a rad trio can be found using the function g(d) =4d-7, where d is the distance traveled. The distance traveled by the same car at a given time, t, can be forgiven he using the function d(t)=2t+4. What is the amount of gas used by a car on a road trip as a function of time?
For k. I meant to put the cubed root of 64 in the parenthesis and for l. I meant to put the fourth root of 3.
(64 to the root of 3)^4
= 4^4
= 256
(3 to the root of 4)^-8
= (3^1/4)^(-8)
= 3^-2
= 1/9
(5^2/5/^2)^-1/2
= 1^-1/2
= 1
or something. bad syntax
#4. Just plug in your two values of r and find the difference.
#2 Just substitute for d
g = 4d-7 = 4(2t+4)-7
To solve each of the given expressions and problems, I'll break them down step by step:
k. (64 to the root of 3)^4:
To solve this expression, you need to calculate the value of the base raised to the power of an exponent. In this case, the base is 64 to the root of 3, and the exponent is 4.
Step 1: Calculate the base raised to the root of 3.
Take the 64 and find its cube root. The cube root of 64 is 4.
Step 2: Raise the result from step 1 to the power of 4.
Now that you have 4 as the base, raise it to the power of 4. 4^4 equals 256.
Therefore, the value of (64 to the root of 3)^4 is 256.
l. (3 to the root of 4)^-8:
Similar to the previous expression, you need to calculate the value of the base raised to the power of an exponent. The base is 3 to the root of 4, and the exponent is -8.
Step 1: Calculate the base raised to the root of 4.
The root of 4 is simply 2. So, 3 to the root of 4 equals 3^2, which is 9.
Step 2: Raise the result from step 1 to the power of -8.
Since -8 is a negative exponent, we can apply the rule that says a^(-n) equals 1 / (a^n). Therefore, 9^(-8) equals 1 / (9^8).
The exact answer would be a fraction, but if you want a decimal approximation, you can evaluate it using a calculator.
f. (5^2/5/^2)^-1/2:
This expression involves using exponentiation and fractions.
Step 1: Simplify the power of 5.
Calculate 5^2 (which is 25) and 5^(-2) (which is 1/25).
Step 2: Apply the powers.
The expression becomes (25 / (1/25))^(-1/2).
Step 3: Evaluate the simplified expression.
The division of 25 by (1/25) is equivalent to multiplying by 25, resulting in 625. Therefore, the expression becomes 625^(-1/2).
Again, this involves a negative exponent, so we can apply the rule a^(-n) = 1 / (a^n). Thus, 625^(-1/2) equals 1 / (625^(1/2)).
You can simplify 625^(1/2) by taking the square root of 625, which is 25.
Therefore, 625^(-1/2) equals 1/25.
4. To find how much the radius of a spherical balloon increases when the amount of air is increased from 4.2ft^3 to 24.9ft^3, we can use the formula r = (0.75V/π)^(1/3), where r is the radius and V is the volume.
Step 1: Calculate the initial radius.
Plug in the initial volume, V1 = 4.2ft^3, into the formula: r1 = (0.75 * 4.2ft^3 / π)^(1/3).
Step 2: Calculate the final radius.
Plug in the final volume, V2 = 24.9ft^3, into the formula: r2 = (0.75 * 24.9ft^3 / π)^(1/3).
Step 3: Find the difference in radii.
Subtract the initial radius (r1) from the final radius (r2): r2 - r1.
The result will give you the increase in radius by how many inches.
2. To find the amount of gas used by a car on a road trip as a function of time, we can use two given functions:
g(d) = 4d - 7 (where d is the distance traveled)
d(t) = 2t + 4 (where t is the time)
Step 1: Express the gas used (g) in terms of time (t).
Substitute the function d(t) into the function g(d) to get g(t):
g(t) = 4(d(t)) - 7.
Step 2: Replace d(t) with its expression in terms of t.
Since d(t) = 2t + 4, we substitute it in the formula: g(t) = 4(2t + 4) - 7.
Step 3: Simplify the equation to get the final function.
Distribute 4 to 2t and 4, then subtract 7: g(t) = 8t + 16 - 7.
Finally, simplify further: g(t) = 8t + 9.
The resulting function g(t) = 8t + 9 represents the amount of gas used by the car on a road trip as a function of time.